Maximal volume entropy rigidity for RCD∗(−(N−1),N) spaces

For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is above $n-1$. If compact, it known that equality holds if and only hyperbolic. We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces. While upper bound straightforward, rigidity case quite involved due lack of a smooth structure in $\mathsf{RCD}^{\ast}$ As an application we obtain almost which partially recovers Cheng-Rong-Xu for manifolds.