Quantitative symmetry bound for universal covers

Let $M$ be a closed Riemannian manifold, let $\widetilde{M}$ be its universal cover, and let $\operatorname{Isom}(\widetilde{M})$ be its isometry group containing $\pi_1(M)$. Quantitative symmetry conjecture. In the characterizations of locally symmetric manifolds, the hypothesis that $\operatorname{Isom}(\widetilde{M})$ is not discrete can be replaced by

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

where $C$ depends only on $\pi_1(M)$. This is the quantitative form of the preceding rigidity and characterization results; the source gives no resolution of the conjecture.