ZFC and ω-resolvability of crowded pseudocompact spaces

Determine whether Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) proves that every crowded pseudocompact topological space is ω-resolvable. Establishing this would, in particular, preclude the existence of crowded Baire pseudocompact Δ-spaces.

Background

The paper shows that if a crowded Baire T1-space is not ω-resolvable, then it contains a crowded Baire irresolvable subspace, whose existence implies an inner model with a measurable cardinal. Since pseudocompact spaces are Baire, the authors consider whether crowded pseudocompact Δ-spaces can exist.

They note that the status of ω-resolvability for crowded pseudocompact spaces is unsettled in ZFC and propose that ZFC should imply ω-resolvability for all such spaces. This conjecture, if true, would rule out the existence of crowded Baire pseudocompact Δ-spaces, aligning with their earlier observations about Baire Δ-spaces being non-ω-resolvable.