Can a headmate have its own headmate?
Indeed they can, apparently. No only that, but a headmate's headmate can also have headmates of their own.
If headmates can have headmates* of their own, and the whole "factive" thing establishes that you can have an actual living person as a headmate, then I believe the logical next question is: Can A have B as headmate, who has C, who has D .... who has A, starting the cycle over again? Infinite recursion depth!
And of course, the obvious question after that is: Does Douglas Hofstadter have himself as a headmate?
*I'm sure the tumblrs have more specific terminology for this, differentiating "having a headmate" in the "system containing individual" and "individual sharing the head with other individual" interpretations, I mean the former in this case. Better formalised as "Can the set A have B as member, where B is a set that has A as a member? Of course, this violates Z-F set theory, but I doubt they would care.
In unrelated commentary:

Why is this here?