This is a more focused version of a question which was asked at MSE a couple years ago, but is still unanswered there.

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 $S\in S$?

I'd also be interested in an answer for other positive set theories; for example, in a comment to the linked question James Hanson suggested that checking whether $S\in S$ in a particular natural model of $\mathsf{PST}+\neg\mathsf{Inf}$ might be feasible, which would at the very least give a consistency result one way or the other. But my primary interest is in $\mathsf{GPK}_\infty^+$ specifically.