Topologically free minimal actions without dynamical comparison

We show the existence of a topologically free minimal action of $\mathbb F_\infty$ on the Cantor space that does not have dynamical comparison.

Moreover, we show that this phenomenon can happen both in the presence and in the absence of invariant measures.

We also show that strict comparison of the reduced crossed product C*-algebra does not imply dynamical comparison for minimal actions.

Our technique involves constructing a monoid which is not almost unperforated, embedding it into a countable refinement monoid and then realising it as the type semigroup associated to a dynamical system.