Mean dimension and radius of comparison

Consider a minimal and free topological dynamical system $(X,\mathrm{\Gamma })$, where $\mathrm{\Gamma }$ is a discrete amenable group. Assume $(X,\mathrm{\Gamma })$ has the uniform Rokhlin property (URP) and the Cuntz comparison of open sets (COS), then it is shown that $$\mathrm{r}\mathrm{c}(\mathrm{C}(X)⋊\mathrm{\Gamma })=\frac{1}{2}\mathrm{m}\mathrm{d}\mathrm{i}\mathrm{m}(X,\mathrm{\Gamma })\mathrm{.}$$ The same statement also holds for unital simple AH algebras with diagonal maps. The arguments in both cases are based on the recent preprint of OpenAI dealing with $\mathbb{Z}$-actions and the topological ideas used in that paper.