Finite decomposition complexity implies finite dynamical complexity for Cantor actions

Let $\Gamma$ be a group with finite decomposition complexity, and let $X$ be the Cantor set. For a free action of $\Gamma$ on $X$, write $X\rtimes\Gamma$ for the associated transformation groupoid. A groupoid has finite dynamical complexity when its dynamical complexity rank is finite.

Cantor-action complexity conjecture. If $\Gamma$ has finite decomposition complexity, then $X\rtimes\Gamma$ has finite dynamical complexity for every free action of $\Gamma$ on the Cantor set.

The conjecture would connect a coarse-geometric finiteness property of the acting group with the complexity of transformation groupoids; the paper presents it as a proposed consequence of techniques for free Cantor actions, rather than as an established result.