Generalizing discreteness of countable subsets of ℋ(X) to higher cardinals

Ascertain whether Proposition 5.2 extends to uncountable cardinals: for a crowded compact zero-dimensional F-space X in which every nonempty Gδ-subset has nonempty interior—particularly for X=ω*—determine whether every subset of the homeomorphism group ℋ(X) of cardinality κ is discrete for cardinals κ with ω<κ<𝔠.

Background

Proposition 5.2 proves that for such spaces X, all countable subsets of ℋ(X) are discrete. For ℋ(ω*), the authors also show that the group is crowded with density 𝔠, implying the existence of non-discrete subsets of size 𝔠; thus the remaining uncertainty concerns intermediate cardinalities between ω and 𝔠.

They further pose a specific related question (without the required explicit marker) asking whether in ZFC there exists a subset of ℋ(ω*) of size ω1 that is not closed.