The entropy rigidity conjecture

Let $M$ be a closed manifold which admits a locally symmetric Riemannian metric $g_{loc}$ with nonpositive sectional curvature. Assume that $(M,g_{loc})$ has no local factors isometric to $\mathbb{R}$. Let $(N,g)$ be any closed Riemannian manifold, and let $f:N\longrightarrow M$ be any continuous map. The normalized entropy is $\operatorname{ent}(g)=h(g)^n\operatorname{Vol}(M,g)$, where $h(g)$ is the volume entropy. Entropy rigidity conjecture.

$$\operatorname{ent}(N,g)\geq |\operatorname{deg} f|\operatorname{ent}(M,g_{loc})$$

with equality if and only if $f$ is homothetic to a Riemannian covering. This extends entropy rigidity from metrics to maps; the metric and topological forms are known in several negatively curved and locally symmetric cases, but the stated generality is not resolved in the supplied source.