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Suitable sets in powers of A(G) for certain non-separable k_ω-group G

Determine whether there exists an integer n ≥ 2 such that the n-th power A(G)^n of the free Abelian topological group over a non-separable k_ω-topological group G that contains no non-trivial convergent sequences has a suitable set.

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Background

The paper proves that under certain assumptions A(X) lacks suitable sets when X is a non-separable k_ω-space without non-trivial convergent sequences. It then raises whether passing to finite powers of A(G) might restore the existence of suitable sets.

This question is explicitly flagged by the authors as unknown and is posed for k_ω-topological groups G (not just general spaces X), focusing on the behavior of A(G)n for some n ≥ 2.

References

However, the following question is unknown for us. Let $G$ be a non-separable $k_{\omega}$-topological group without non-trivial convergent sequences. Does $A(G){n}$ have a suitable set for some $n\geq 2$?

— Suitable sets for topological groups revisited (2508.13443 - Lin et al., 19 Aug 2025) in Question \ref{q0}, Section 3 (Free topological groups with a suitable set)