o-Boundedness of free topological groups

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Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle Ufin(O,Ω) (the latter means that for every sequence <uₙ>ₙ∈ w of open covers of T there exists a sequence <vₙ>ₙ∈ w such that vₙ∈ [uₙ]<ʷ and for every F∈ [X]<ʷ there exists n∈ w with F⊂∪ vₙ). This characterization gives a consistent answer to a problem posed by C. Hernandes, D. Robbie, and M. Tkachenko in 2000.