John, as you quoted in your post, Form S-1 "must be filed before shares are listed on a national Exchange", but it's not the only case for filing it.
I'm expecting the company filing it mainly because it was mentioned in the first SC 13D filing and that part of the registration rights agreement was not ammended with the new SC 13D/A, just dates and other details:
(a) Mandatory Registration . The Company shall prepare and, as soon as practicable, but in no event later than the Filing Deadline, file with the SEC an initial Registration Statement on Form S-1 (or Form S-3, if available for use at the time of filing) covering the resale of all of the Registrable Securities, provided that such initial Registration Statement shall register for resale at least the number of shares of Common Stock equal to the Required Registration Amount as of the date such Registration Statement is initially filed with the SEC. Such initial Registration Statement, and each other Registration Statement required to be filed pursuant to the terms of this Agreement, shall contain (except if otherwise directed by the Required Holders) the “ Selling Stockholders ” and “ Plan of Distribution ” sections in substantially the form attached hereto as Exhibit A . The Company shall use its best efforts to have such initial Registration Statement, and each other Registration Statement required to be filed pursuant to the terms of this Agreement, declared effective by the SEC as soon as practicable, but in no event later than the applicable Effectiveness Deadline for such Registration Statement.
Also note the following from the AMENDMENT NO. 1 TO REGISTRATION RIGHTS AGREEMENT:
AMENDMENT NO. 1 TO REGISTRATION RIGHTS AGREEMENT
This Amendment (the “Amendment”) to the Registration Rights Agreement (as defined herein) is dated as of May 30, 2019 (the “Amendment Effective Date”), by and between Verus International, Inc., a Delaware corporation (the “Company”) and ARJ Consulting LLC, a New York limited liability company (the “Buyer”). WHEREAS, on February 8, 2019, the Company and the Buyer entered into a registration rights agreement (the “Registration Rights Agreement”); and WHEREAS, the Company and the Buyer desire to make certain amendments to the Registration Rights Agreement as set forth herein. NOW THEREFORE, in consideration of the above, and for good and valuable consideration, the receipt and sufficiency of which is hereby acknowledged, the parties hereto agree as follows: 1. Any capitalized but undefined terms referenced herein shall bear such meaning ascribed to them in the Registration Rights Agreement.