Farb and Weinberger's characterization of locally symmetric manifolds

Let $M$ be a closed Riemannian $n$-manifold with $n>1$. Write $\widetilde{M}$ for its universal cover, $\operatorname{Isom}(\widetilde{M})$ for its isometry group, and let $\pi_1(M)$ act on $\widetilde{M}$ by deck transformations. A manifold is smoothly irreducible if it is not finitely covered by a product. There exists a constant $C$ depending only on $\pi_1(M)$ such that the following are equivalent:

Farb and Weinberger's conjecture.

1. $M$ is aspherical, smoothly irreducible, $\pi_1(M)$ has no non-trivial normal abelian subgroup, and

$$[\operatorname{Isom}(\widetilde{M}):\pi_1(M)]>C.$$

2. $M$ is isometric to an irreducible, locally symmetric Riemannian manifold of non-positive sectional curvature.

This conjecture characterizes the manifolds with sufficiently large symmetry degree among closed aspherical manifolds satisfying the stated irreducibility and fundamental-group hypotheses. The surrounding results establish bounded symmetry for closed piecewise locally symmetric manifolds and their existence of minimal orbifolds, while the characterization attributed to Farb and Weinberger remains unresolved here.