Gromov's average area ratio conjecture for hyperbolic manifolds

Let $(M,h_0)$ be a closed hyperbolic manifold of dimension $n\geq 3$, and let $N$ be a closed $n$-manifold with Riemannian metric $g$ whose scalar curvature satisfies $R_g\geq -n(n-1)$. Let $F:N\to M$ be a smooth map of degree $d$. Gromov's conjecture. The average area ratio satisfies

$$\operatorname{Area}_F(g/h_0)\geq d.$$

Moreover, equality holds if and only if $F$ is a local isometry. This conjecture strengthens Gromov's known lower bound in dimension three from $d/3$ to $d$ and proposes the analogous sharp inequality in every dimension $n\geq 3$; the supplied context does not state whether it has been resolved.