The PFA(S)[S] conjecture on first-countable perfect pre-images of $\omega_1$

Let $X$ be a first-countable perfect pre-image of $\omega_1$, and let $\mathrm{PFA}(S)[S]$ denote the stated forcing axiom and its associated extension.

PFA(S)[S] conjecture. Under $\mathrm{PFA}(S)[S]$, every first-countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.

The conjecture would sharpen the paper's consistency results involving hereditary normality by allowing “copy of $\omega_1$” to replace “perfect pre-image of $\omega_1$.” The source also mentions an unpublished theorem of the author related to this claim, but the supplied text gives no resolution of the conjecture.