Equivalence of Sofic $p$-Metric Mean Dimensions and a Tame-Metric Variational Formula

Let $\Gamma$ be a countable discrete sofic group acting by homeomorphisms on a compact metrizable space $X$ and $\Sigma$ a sofic approximation of $\Gamma$. We prove that for every $1\leq p<\infty$, the sofic $p$-metric mean dimension is equivalent to the sofic metric mean dimension, i.e there exists a common value $ D_\Sigma(X,\Gamma)\in\{-\infty\}\cup[0,+\infty]$ such that, $$D_\Sigma(X,\Gamma)=\mdim_{\Sigma,\mathrm M,p}(X,\Gamma)
=\mdim_{\Sigma,\mathrm M,\infty}(X,\Gamma),$$
which answers a question of Hayes in \cite[Question 3]{Hayes}.
Moreover, a tame-metric variational formula is established. That is for every $1\leq q\leq\infty$, $$D_\Sigma(X,\Gamma)
=\inf_{\rho\in\mathcal T(X)}
\underline{\mdim}_{\Sigma,q}(X,\rho), $$
where $\mathcal T(X)$ is the set of all compatible metrics on $X$ having tame growth of covering numbers.