Z-stability for nonamenable actions

Prove that, for every amenable, topologically free, minimal action of a nonamenable group on a compact metrizable space, the crossed product is Z-stable.

Background

For amenable actions of nonamenable groups, nuclearity and simplicity can hold while the acting group itself is nonamenable. The survey states that Z-stability is expected to be automatic in this setting, but presents the assertion as an unresolved conjecture.