Suitable sets in free Abelian topological groups over non-separable compact spaces without convergent sequences

Determine whether, for every non-separable compact Tychonoff space X that contains no non-trivial convergent sequences, the free Abelian topological group A(X) does not belong to the class S of topological groups that have a suitable set; that is, establish whether A(X) has no suitable set for all such X.

Background

A suitable set for a topological group G is a discrete subset S such that S ∪ {e} is closed in G and the subgroup generated by S is dense in G. The class S denotes all groups that have such a suitable set. The problem asks whether the free Abelian topological group A(X) over a compact, non-separable space X that has no non-trivial convergent sequences necessarily lacks any suitable set.

This question traces back to Problem 2.1 in Tkachenko (1997). The present paper studies existence of suitable sets in various contexts and provides partial answers related to this problem (e.g., Theorem tttt and its corollaries).