Existence of a finitely generated group with infinite Gaschütz rank

Determine whether there exists a finitely generated discrete group with infinite Gaschütz rank; equivalently, ascertain whether the free group F_n on n generators has finite Gaschütz rank for every n.

Background

The paper shows that connected Lie groups always have finite Gaschütz rank, but it remains unsettled whether this finiteness extends to finitely generated discrete groups.

The authors note that this question is equivalent to asking whether all free groups F_n have finite Gaschütz rank, reflecting a central theme connecting generator lifting phenomena in discrete and profinite settings.