The "yours to keep" message can show up in cases where there is no link with rel="open-access". This was reported against a feed from ProQuest with OPDS that looks like this:

 <link href="https://nyu-test.se.lyrasistechnology.org/NYU/works/ProQuest%20Doc%20ID/5205544/borrow" rel="http://opds-spec.org/acquisition/borrow" type="application/atom+xml;type=entry;profile=opds-catalog">
<opds:indirectAcquisition type="application/epub+zip"/>
<opds:availability status="available"/>
</link>

Here's how that OPDS is rendered on iOS SimplyE: 

This doesn't happen on iOS for other books with superficially similar OPDS – search for "The Moon Rock" in NYPL's collection for an example.

Even if we change nothing else, the "open-access" message should only be displayed when there is a link with rel="open-access".

I don't know for sure whether this problem happens on Android – we can't render a real ProQuest feed on Android because we don't yet have SAML authentication support. Looking at the source code, it seems unlikely that it has the exact same problem, but we should check.

More information can be found in SIMPLY-3356's comments section.