Volume rigidity conjecture for closed hyperbolic manifolds

Let $(M^n,g_H)$ be a closed hyperbolic $n$-manifold with $n\geq 5$, and let $g$ be a Riemannian metric on $M$ satisfying

$$\operatorname{Sc}_g=-n(n-1),\qquad \operatorname{Vol}_g(M^n)=\operatorname{Vol}_{g_H}(M^n).$$

Volume rigidity conjecture. The metric $g$ is isometric to the hyperbolic metric $g_H$.

This conjecture proposes rigidity at fixed negative scalar curvature and hyperbolic volume in dimensions at least five. The source provides no resolution of the conjecture.