Suitable sets in subgroups of G^{\star}(ω) when τ = ω₁

Determine whether every subgroup of G^{\star}(ω) has a suitable set when τ = ω₁, where G^{\star}(ω) denotes the subgroup of the product ∏_{α<τ} G_α consisting of elements with countable support, equipped with the subspace topology inherited from the linear group topology G^{\star}.

Background

The group G^{\star} is the product ∏{α<τ} Gα endowed with a linear topology having a base of clopen subgroups U_α = ∏{α≤β<τ} Gβ. The subgroup G^{\star}(ω) consists of elements with countable support and carries the subspace topology of G^{\star}.

The authors establish existence of suitable sets for subgroups of G^{\star}(ω) in several cardinal cases (τ = ω and τ > ω₁) and explicitly single out τ = ω₁ as a remaining unknown.