Topologically free actions of vigorous groups

Determine whether every vigorous group admits a minimal, extremely proximal, and topologically free action on a Hausdorff space.

Background

A vigorous group is defined through a faithful piecewise minimal-extremely-proximal action on the Cantor set, and its canonical action is never topologically free. Ozawa's theorem establishes C*-selflessness for groups admitting a minimal, extremely proximal, and topologically free action on a Hausdorff space.

The paper notes that it is unknown whether every vigorous group admits such an alternative action. This unresolved issue explains why Ozawa's theorem cannot directly be applied to vigorous groups in general; the paper instead proves their C*-selflessness by using averaging inside rigid stabilizers and free subgroups.