Minimal amenable subshift with full mean dimension

Let $G$ be an infinite countable amenable group and $P$ a polyhedron with topological dimension $dim(P)<\mathrm{\infty }$. We construct a minimal subshift $(X,G)$ such that its mean topological dimension is equal to $dim(P)$. This result answers the question of D. Dou in \cite{DD}, moreover, it is also an extension of the work of L. Jin and Y. Qiao \cite{JQ} for $\mathbb{Z}$-action.