Relative uniformly positive entropy of induced amenable group actions

Let $G$ be a countable infinite discrete amenable group.It should be noted that a $G$-system $(X,G)$ naturally induces a $G$-system $(\mathcal{M}(X),G)$, where $\mathcal{M}(X)$ denotes the space of Borel probability measures on the compact metric space $X$ endowed with the weak*-topology. A factor map $\pi \text{\hspace{0.17em}⁣}:(X,G)\to (Y,G)$ between two $G$-systems induces a factor map $\tilde{\pi }\text{\hspace{0.17em}⁣}:(\mathcal{M}(X),G)\to (\mathcal{M}(Y),G)$. It turns out that $\tilde{\pi }$ is open if and only if $\pi$ is open. When $Y$ is fully supported, it is shown that $\pi$ has relative uniformly positive entropy if and only if $\tilde{\pi }$ has relative uniformly positive entropy.