This is a more focused version of a question which was asked at MSE a couple years ago, but is still unanswered there. That question asks about a broad range of theories, whereas this version focuses on a single one.

Let $S=\{x: x\in x\}$ be the "dual" to Russell's paradoxical set $R$, and consider the question "Is $S\in S$?" If we try to work purely "naively" there does not seem to be an immediate argument one way or the other (dually to the situation for $R$). However, in light of Lob's theorem it's a bit premature to leap to the conclusion that there actually are no such arguments.

So it remains plausible that there is a "reasonable" set theory in which the question "Is $S\in S$?" is decided in a nontrivial way. Note that the nontriviality requirement rules out both $\mathsf{ZF}$- and $\mathsf{NF}$-style set theories: the former prove $S=\emptyset$, while the latter prove that $S$ is not a set in the first place.

The most appealing candidate to me appears to be $\mathsf{GPK}_\infty^+$. This theory proves that $S$ is a set and is nonempty. It also (in my opinion) has a very attractive intuition behind it, as well as an interesting model theory. So my question is:

Does $\mathsf{GPK}_\infty^+$ decide whether $S\in S$?