Monotonicity of the n-Gaschütz property

Determine whether, for a continuous epimorphism f: G → H between topological groups, the property of being n-Gaschütz implies that f is also (n+1)-Gaschütz.

Background

Definition 1.4 introduces the notion of an epimorphism f: G → H being n-Gaschütz, meaning that every n-element generating set of H lifts to a generating n-tuple of G. The authors point out that it is not clear whether increasing the number of generators by one preserves the lifting property for general topological groups.

This question concerns a basic monotonicity property of lifting generators and ties directly to understanding the Gaschütz rank boundaries introduced in the paper.