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The PFA(S)[S] conjecture on first-countable perfect pre-images of $\omega_1$
About 15 years old
PFA(S)[S] conjecture. Under $\mathrm{PFA}(S)[S]$ , every first-countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$ .
openforcing-axioms locally-compact-spaces set-theoretic-topology
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The PFA(S)[S] conjecture on first-countable perfect pre-images of ω1\omega_1ω1​