Quantitative maximal volume entropy rigidity on Alexandrov spaces

We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given $N, D$, there exists $epsilon(N, D)>0$, such that for $epsilon<epsilon(N, D)$, if $X$ is an $N$-dimensional Alexandrov space with curvature $geq -1$, $operatorname{diam}(X)leq D, h(X)geq N-1-epsilon$, then $X$ is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity of cite{CRX} to Alexandrov spaces.