Almost finiteness of minimal essentially free elementary amenable group actions

Let $G$ be an elementary amenable group acting minimally and essentially freely on an infinite compact metrizable space $X$ with the small boundary property. The action is almost finite if it satisfies the corresponding dynamical tower-decomposition condition, and its crossed product is $C(X)\rtimes_r G$. Almost finiteness conjecture. Every such action is almost finite, and the crossed product is $\mathcal{Z}$-stable. This would extend known almost-finiteness and regularity results beyond the virtually $\mathbb{Z}$ setting and would imply strong structural properties for the associated crossed products; the conjecture is presented as open in the source.