EXHIBIT 10.31

Exhibit 10.31

Lake Street Capital Markets, LLC

121 South 8th Street, Suite 1000

Minneapolis, Minnesota 55402

Laidlaw & Company (UK) Ltd.

521 Fifth Avenue, 12th Floor

New York, NY 10175

July [●], 2026

Private and Confidential

AEON Biopharma, Inc.

5 Park Plaza

Suite 1750

Irvine, CA 92614

Attention:                Robert Bancroft

President and Chief Executive Officer

Dear Robert:

We understand that AEON Biopharma, Inc., a Delaware corporation (the “Company”), has requested that Lake Street Capital Markets, LLC and Laidlaw & Company (UK) Ltd. (collectively, the “Agents”) act as the Company’s solicitation agents in connection with the exercise of the Common Warrants (as defined in the Underwriting Agreement) and the Option Common Warrants (as defined in the Underwriting Agreement) (the Common Warrants and the Option Common Warrants together, the “Milestone Warrants”) to be issued pursuant to that certain Underwriting Agreement, dated as of the date hereof (the “Underwriting Agreement”), between the Company and Lake Street Capital Markets, LLC, as representative of the several underwriters named therein, including Laidlaw & Company (UK) Ltd. (the “Services”). The Agents have agreed to perform the Services subject to the execution of this agreement (the “Letter Agreement”), and the Company hereby authorizes the Agents to contact Warrant holders (the “Holders”) in performing the Services.

As compensation for the Services, no later than five (5) business days following each calendar quarter, the Company shall indicate how many Milestone Warrant Securities have been issued to warrantholders upon the cash exercise of the Milestone Warrants and pay the Agents a fee consisting of a cash payment equal to the Applicable Percentage (as defined below) of the cash exercise payment received by the Company for all cash exercises of the Milestone Warrants during the then-ended calendar quarter (the “Warrant Solicitation Fee”), with such Warrant Solicitation Fee allocated among the Agents in proportion to the number of Firm Securities set forth opposite their respective names in Schedule 1 of the Underwriting Agreement and payable by wire transfer of immediately available funds to a bank account designated by such Agent. The Applicable Percentage shall equal (i) 3.0% if at the time of such cash exercise the aggregate number of Milestone Warrant Securities that have been issued to warrantholders upon the cash exercise of the Milestone Warrants, including the Milestone Warrant Securities issued upon such exercise, equals between 0-[●]1 (subject to appropriate adjustment for stock splits, stock dividends, combinations, and other recapitalizations), (ii) 3.75% if at the time of such cash exercise the aggregate number of Milestone Warrant Securities issued to warrantholders upon the cash exercise of the Milestone Warrants, including the Milestone Warrant Securities issued upon such exercise, equals between [●]-[●]2 (subject to appropriate adjustment for stock splits, stock dividends, combinations, and other recapitalizations) and (iii) 4.50% if at the time of such cash exercise the aggregate number of Milestone Warrant Securities issued to warrantholders upon the cash exercise of the Milestone Warrants, including the Milestone Warrant Securities issued upon such exercise, equals between [●]-[●]3 (subject to appropriate adjustment for stock splits, stock dividends, combinations, and other recapitalizations). If upon the cash exercise of a Milestone Warrant a portion of the Milestone Warrant Securities subject to exercise falls within more than one of the above-listed Applicable Percentages then the Applicable Percentage shall be pro rata split between such Applicable Percentages (for example, if upon the cash exercise of a Milestone Warrant for 1,000 Milestone Warrant Securities, the first 250 Milestone Warrant Securities were within the Milestone Warrant Security threshold in the preceding clause (i), and the 251st Warrant Share resulted in the aggregate Milestone Warrant Securities qualifying for the Milestone Warrant Securities threshold in the preceding clause (ii), then the Applicable Percentage would equal 3.00% as it relates to the first 250 Milestone Warrant Securities and 3.75% as it relates to the remaining 750 Milestone Warrant Securities). “Milestone Warrant Securities” means the shares of Class A common stock, par value $0.0001 per share, of the Company (the “Common Stock”), issued upon the exercise of the Milestone Warrants and the shares of Common Stock underlying any Pre-Funded Warrants (as defined in the Underwriting Agreement) issued upon exercise of the Milestone Warrants.

1 NTD: To equal 24.99% of the shares of common stock underlying the Milestone Warrants.

2 NTD: To equal 25.00%-49.99% of the shares of common stock underlying the Milestone Warrants.

3 NTD: To equal 50.00%-100% of the shares of common stock underlying the Milestone Warrants.

Upon the request of one or more Agents, the Company will furnish the Agents with all information concerning the Company and the Holders that the Agents reasonably deem appropriate and necessary for the purpose of rendering the Services. The Company understands and confirms that (i) the Agents will rely, without independent investigation or verification, upon the accuracy and completeness of such information supplied by or on behalf of the Company and upon publicly available information and such other information as the Agents may each deem appropriate and necessary in connection with the Services, and (ii) the Agents will not be responsible for assuming, and do not assume, any responsibility for independent investigation or verification of any such information or for the accuracy or completeness thereof, and shall not be liable for any inaccuracies or omissions therein. The Company will notify the Agents promptly of (A) any material inaccuracy or misstatement in, or material omission from, any information previously furnished to the Agents and (B) the receipt by the Company of any communication from the SEC or any state securities commissioner or any regulatory authority or the commencement of any lawsuit or proceeding to which the Company is a party relating to the Milestone Warrants.

The Company acknowledges and agrees that none of the Agents, any of their respective affiliates or their respective officers, directors, employees, agents and controlling persons (each, an “Advisor Party”) shall have any liability (whether direct or indirect, in contract, tort or otherwise) to the Company or any other person asserting claims on behalf of or in right of the Company for any Loss (as defined below) (a) by reason of any unauthorized use, reliance, publication, distribution of or reference to any of the advice or opinions provided by the Agents or any unauthorized reference to the Agents or the engagement or (b) in connection with the Services, the engagement or the Agents’ services performed or contemplated thereby, except in the case of clause (b), solely with respect to the Company, to the extent that any such Loss is finally judicially determined by a court of competent jurisdiction to have resulted from such Advisor Party’s gross negligence or willful misconduct in the performance of the Services.

In addition, the Company (the “Indemnifying Party”) agrees to indemnify and hold harmless each of the Agents, each of their affiliates and their and each of their respective officers, directors, employees, advisors, representatives, agents and controlling persons (each of the foregoing, an “Indemnified Party” and collectively, the “Indemnified Parties”) from and against any and all losses, claims, demands, damages, liabilities and expenses of any kind (including, without limitation, legal fees and other expenses incurred in connection with any suit, action, investigation or proceeding or any claim asserted), joint or several (collectively, “Losses”), incurred by any of the Indemnified Parties or to which any such Indemnified Party may become subject:

The Indemnifying Party further agrees to reimburse immediately upon request such Indemnified Parties for any expenses (including, without limitation, legal fees and other expenses incurred in connection with any suit, action, investigation or proceeding or any claim asserted) as they are incurred in connection with investigating, preparing for and defending or providing evidence in any of the foregoing, including in connection with the enforcement of the indemnification and reimbursement provided for herein. Notwithstanding the foregoing, in the case of clause (ii) of the preceding paragraph, the Indemnifying Party shall not have any indemnification or reimbursement obligation to any Indemnified Party to the extent that any Loss is finally judicially determined by a court of competent jurisdiction to have resulted solely from the gross negligence or willful misconduct of such Indemnified Party in the performance of its services hereunder.

For the avoidance of doubt, the gross negligence or willful misconduct of any Agent shall not be attributed to or otherwise affect the indemnification, exculpation or any other rights hereunder of the other Agent, and no Agent shall be a representative or agent of the other.

If for any reason the foregoing indemnification and reimbursement is unavailable to any Indemnified Party or insufficient in respect of any Losses referred to therein to hold it harmless, then the Indemnifying Party agrees to contribute to the amount paid or payable by such Indemnified Party as a result of such Losses (i) in such proportion as is appropriate to reflect the relative benefits received by the Indemnifying Party, on the one hand, and such Indemnified Party, on the other hand, of the Services or (ii) if (but only if) the allocation provided for in clause (i) is for any reason unenforceable, in such proportion as is appropriate to reflect not only such relative benefits but also the relative fault of the Indemnifying Party, on the one hand, and such Indemnified Party, on the other hand, as well as any relevant equitable considerations. It is hereby agreed that for purposes of this paragraph the relative benefits to the Indemnifying Party, on the one hand, and all Indemnified Parties, on the other hand, of the Services shall be deemed to be in the same proportion as (i) the gross aggregate cash from the exercise of the Milestone Warrants (before deducting expenses) received by the Company, bears to (ii) the total Warrant Solicitation Fee received by the Agents,; provided that in no event shall the Indemnified Parties be required to contribute an aggregate amount in excess of the fees actually received by such Agent. The parties agree that it would not be just and equitable if contribution were determined by pro rata allocation or by any other method of allocation that does not take into account the equitable considerations referred to above.

Promptly after receipt of notice of the commencement of any Proceeding against the Agents or any other Indemnified Party, the Agents or any other Indemnified Party shall notify the Indemnifying Party in writing of the commencement thereof; provided that any failure or delay in so notifying shall not relieve the Indemnifying Party of any liability that it may have to the Agents or any other Indemnified Party except and only to the extent that such failure materially prejudices the ability to defend the Proceeding. The Indemnifying Party shall have 30 days after receipt of notice to undertake, at its own expense, the settlement or defense of the Proceeding, including prompt employment of counsel acceptable to the Indemnified Parties affected and payment of all expenses. The relevant Indemnified Parties shall have the right to participate in the settlement or defense of the Proceeding. The relevant Indemnified Parties may retain one firm of separate counsel in each relevant jurisdiction to represent them in the settlement or defense of the Proceeding, at the expense of the Indemnifying Party, if (i) the Indemnifying Party does not assume the defense of the Proceeding within 30 days of receipt of notice; (ii) the Indemnifying Party agrees to separate representation; (iii) the Agents are advised in writing by counsel that there is an actual or potential conflict in the Indemnifying Party’s and the Indemnified Parties’ respective interests or additional defenses are available to the Indemnified Parties that are not available to the Indemnifying Party, which makes representation by the same counsel inappropriate or (iv) the Agents are advised in writing by counsel that the subject matter of the Proceeding may not fall within the indemnity set forth herein, which makes representation by the same counsel inappropriate. The Indemnifying Party agrees that, without an Indemnified Party’s prior written consent, it will not settle, compromise or consent to the entry of any judgment in or otherwise seek to terminate any Proceeding in respect of which indemnification could be sought under the indemnification provisions of this Letter Agreement (whether or not the Agents or any other Indemnified Party is an actual or potential party to such Proceeding), or participate in or otherwise facilitate any such settlement, compromise, consent or termination by or on behalf of any person or entity, unless such settlement, compromise, consent or termination (i) includes an unconditional release of such Indemnified Party, in form and substance satisfactory to such Indemnified Party, from, and holds such Indemnified Party harmless against, all liability arising out of such Proceeding and (ii) does not include any statement as to or any admission of fault, culpability or a failure to act by or on behalf of any Indemnified Party.

The Indemnifying Party will promptly notify an Indemnified Party of the assertion against it or any other person of any Proceeding or the commencement thereof relating to or arising out of any matter referred to in this Letter Agreement, including, without limitation, an Indemnified Party’s services hereunder.

To help the government fight the funding of terrorism and money laundering activities, U.S. federal law requires financial institutions to obtain, verify and record information that identifies each individual or institution that opens an account or establishes a new client relationship with the Agents. Accordingly, pursuant to Section 326 of the USA PATRIOT Act, 31 U.S.C. §5318, the Agents may request information, including the individual or institution’s full legal name, address, taxpayer identification number and other information to verify such person’s identity as part of the Agents' AML compliance program. If all required documentation or information is not provided, the Agents may be unable to open an account or establish a relationship with the Company. This section hereby notifies the Company of such requirement.

None of the Company, any of its subsidiaries, directors or officers or, to the best knowledge of the Company, any of its affiliates, agents or employees, has engaged in any activity or conduct that would violate any applicable anti-bribery, anti-corruption or anti-money laundering laws or regulations in any applicable jurisdiction and the Company has instituted and maintains policies and procedures designed to prevent violation of such laws, regulations and rules. None of the Company, any of its subsidiaries, directors or officers or, to the best knowledge of the Company, any of its affiliates, agents or employees, is currently a person with whom dealings are restricted or prohibited by any Sanctions (as defined below). The Company covenants and agrees that it will not directly or indirectly use the proceeds from the exercise of the Milestone Warrants or lend, contribute or otherwise make available such proceeds to any subsidiary, sister company, joint venture partner or other person or entity (i) for the purpose of financing or facilitating any activity of or transaction or business with any person or entity, or in any country or territory, that, at the time of such funding or facilitation, is a person, entity, country or territory with which dealings are restricted or prohibited by any Sanctions (including any person owned or controlled by any person subject to any Sanctions) or (ii) in any other manner that will result in a violation by any person (including any person participating in the Services) of Sanctions. For the purpose of this Letter Agreement, “Sanctions” means any economic or trade sanctions or restrictive measures enacted, administered, imposed or enforced by the U.S. Department of the Treasury’s Office of Foreign Assets Control (OFAC), the U.S. Department of State, the United Nations Security Council, His Majesty’s Treasury and/or the European Union or other relevant sanctions authority.

The term of this Letter Agreement shall extend from the date hereof until the date on which all of the Milestone Warrants have expired by their terms (the “Term”). Each Agent, with respect to itself, may terminate this Letter Agreement at any time, with or without cause, by giving the Company at least five calendar days’ prior written notice. If this Letter Agreement is terminated by one or more Agents, this Letter Agreement shall thereafter have no effect with respect to any such terminating Agent except the allocation of the Warrant Solicitation Fee shall be proportionally adjusted such that the continuing Agent will each receive the percentage of the Warrant Solicitation Fee previously allocated to the departing Agent. The indemnification obligations herein shall survive the termination or expiration of this Letter Agreement for Losses occurring prior to the date of such termination or expiration.

Each Agent (i) has been retained hereunder solely to provide the Services as provided for herein, and not as an advisor to any other person, and any duties (which will not be fiduciary duties) arising out of the Agents’ engagement will be owed solely to the Company with respect to the matters contemplated hereby and not to any other person, (ii) is engaged hereunder as an independent contractor, (iii) is not acting as an agent of the Company or in a fiduciary capacity with respect to the Company or any other person, and (iv) is not and will not be responsible for any underlying business decision to effect the Services.

Notwithstanding anything herein to the contrary, the Company shall be permitted, without the prior written consent of the Agents, to (i) describe this Letter Agreement (including the Warrant Solicitation Fee and the material terms hereof) in, and file this Letter Agreement as an exhibit to, the Registration Statement, any final prospectus, any Current Report on Form 8-K, or any other filing required or advisable to be made with the SEC or FINRA in connection with the Offering or the Milestone Warrants, (ii) disclose the existence and terms of this Letter Agreement to its directors, officers, employees, legal counsel, auditors, financial advisors and other professional advisors on a need-to-know basis, and (iii) disclose the existence and terms of this Letter Agreement as required by law, the rules of any applicable securities exchange, or any subpoena or similar legal process. Except as expressly permitted by the foregoing, the Company agrees not to disclose the existence of this Letter Agreement or the contents hereof without the prior written consent of the Agents (such consent not to be unreasonably withheld, conditioned or delayed).

Notice given pursuant to any of the provisions of this Letter Agreement shall be in writing and shall be mailed, emailed or delivered (a) to the Company, at the address listed on the front of this Letter Agreement, and at the email address: aw@aeonbiopharma.com; (b) Lake Street Capital Markets, LLC, 121 South 8th Street, Suite 1000, Minneapolis, Minnesota 55402, Attention: Head of Investment Banking, with a copy sent via email to CapitalMarkets@lakestreetcm.com and to jon.zimmerman@faegredrinker.com; and [(c) 521 Fifth Avenue, 12th Floor, New York, NY 10175, Attention:      Hugh Regan, with a copy sent via email to hregan@laidlawltd.com]4. Notice by mail or delivery shall be deemed conclusively to have been given and received upon delivery to a responsible officer or employee of the addressee, and notice by email shall be deemed conclusively to have been given upon confirmation of receipt by the addressee by sending an email to the sender. Either party may change its address by notice to the other in the manner aforesaid.

This Letter Agreement shall be governed by the laws of the State of New York and shall remain operative and in full force and effect from the date hereof. This Letter Agreement shall be binding upon and inure to the benefit of any successors, assigns, heirs or representatives of the Company or any Indemnified Party. Any right of trial by jury in any action, proceeding or counterclaim (whether based upon contract, tort or otherwise) related to or arising out of this Letter Agreement is hereby waived by the parties hereto. The parties hereto hereby submit to the exclusive jurisdiction of the federal and New York state courts located in the Borough of Manhattan in connection with any dispute related to this Letter Agreement or any of the matters contemplated hereby.

4 NTD: To be confirmed.

This Letter Agreement may be amended, supplemented or changed, and any provision hereof or thereof may be waived, only by a written instrument making specific reference to this Letter Agreement which is signed by the parties hereto. The waiver by any party hereto of a breach of any provision of this Letter Agreement shall not operate or be construed as a further or continuing waiver of such breach or as a waiver of any other or subsequent breach.

This Letter Agreement may be executed in any number of counterparts, each of which shall be deemed to be an original, but all such counterparts shall together constitute one and the same Agreement. Electronic signatures complying with the New York Electronic Signatures and Records Act (N.Y. State Tech. §§ 301- 309), as amended from time to time, or other applicable law will be deemed original signatures for purposes of this Letter Agreement. Transmission by telecopy, electronic mail or other transmission method of an executed counterpart of this Letter Agreement will constitute due and sufficient delivery of such counterpart.