I would have thought that maintaining your HHonors diamond status was based upon holding the Aspire card..
That's what it says in the terms:
Hilton Honors Complimentary Diamond Status
You will receive complimentary Hilton Honors Diamond status with your Hilton Honors American Express Aspire Card. Complimentary Diamond status benefit is only available to the Basic Card Member. If your Hilton Honors American Express Aspire Card is cancelled for any reason, your complimentary Hilton Honors Diamond status provided with the Card will be cancelled. American Express reserves the right to change, modify or revoke complimentary Diamond status at any time. If your complimentary Diamond status is cancelled, you may be able to maintain your Diamond status by qualifying under the Hilton Honors Program terms. Diamond status benefits are subject to availability and vary by hotel. For more information on Diamond status benefits or how to maintain Diamond status each year with qualifying stays, nights or Hilton Honors Base Points through the Hilton Honors Program, and for complete Terms and Conditions, visit HiltonHonors.com/MemberBenefits.
However, American Express practice frequently differs from published policy, and a downgrade may not be detected as a cancelation by the automated systems. I do not think there would be grounds to complain if the status were revoked, but it sounds as if there is a good chance that Hilton will be liberal while their properties are empty.