The almost finiteness conjecture for free amenable group actions

Let $G$ be a countably infinite amenable discrete group acting freely on a finite-dimensional compact metrizable space $X$. Almost finiteness conjecture. The action of $G$ on $X$ is almost finite. Almost finiteness is a central condition in the study of crossed products arising from amenable group actions; it implies regularity properties such as ${\mathcal Z}$-stability and is known in many important examples, but the general finite-dimensional case remains open.