The uniqueness and nonexistence conjecture for Einstein metrics on branched hyperbolic coverings

Let $M$ be a closed hyperbolic manifold of dimension $n\geq 4$, and let $X$ be a nontrivial finite covering of $M$ branched along a closed totally geodesic submanifold of codimension two. An Einstein metric on $X$ is a metric $h$ satisfying ${\rm Ric}_h=ch$ for some constant $c\in\mathbb{R}$. Einstein metric conjecture. The manifold $X$ admits a unique Einstein metric up to scaling, and it does not admit a metric of constant curvature. This conjecture concerns distinguished metrics on branched coverings of hyperbolic manifolds; the source notes that part of it was previously phrased as a question by Gromov and Thurston. The supplied text does not establish the conjecture or give evidence that it has been resolved.