[SalesForce] Unable to delete ContentDocumentLink

This is all on a Salesforce Community.

I have a situation where a user is uploading a file and it's temporarily assigned to their user record. Once they submit the form and it creates a custom object record, I need to re-assign those files to the new custom object record. This is not part of a trigger but a controller behind a custom lightning component.

Here's the code that's doing the work:

List<ContentDocumentLink> oldLinks = [select Id, ContentDocumentId, LinkedEntityId, ShareType from ContentDocumentLink where LinkedEntityId=:userId and ContentDocumentId=:ids];
List<ContentDocumentLink> newLinks = new List<ContentDocumentLink>();

for (ContentDocumentLink old : oldLinks) {
    newlinks.add(new ContentDocumentLink(ContentDocumentId=old.ContentDocumentId, LinkedEntityId=wrapper.objProject.Id, ShareType=old.ShareType));
}

delete oldLinks;
insert newLinks;

The resulting error is:

System.DmlException: Delete failed. First exception on row 0 with id 06Aq0000002COEuEAO; first error: FIELD_INTEGRITY_EXCEPTION, field integrity exception: []

I've now tried to bust this down to a super basic level to just:

List<ContentDocumentLink> toDelete = [select Id from ContentDocumentLink where Id='06Aq0000002COEuEAO'];

delete toDelete;

But I receive the same error. What I can't determine is why this is an issue. That is the only link that exists for that document.

Anyone have any ideas?