Guth's macroscopic Schoen conjecture for hyperbolic manifolds

Let $M$ be a closed hyperbolic manifold, let $g$ be an auxiliary Riemannian metric on $M$, and let $V_{hyp}(R)$ denote the volume of a geodesic ball of radius $R$ in hyperbolic space. Write $\widetilde{M}$ for the universal cover, equipped with the lifted metric $\widetilde{g}$, and let $\widetilde{B}(y,R)\subset\widetilde{M}$ be the metric ball of radius $R$ centered at $y$. Its Riemannian volume is denoted by $|\widetilde{B}(y,R)|$. Guth's conjecture. If

$$\operatorname{vol}(M,g)<\operatorname{vol}(M,hyp),$$

then for every radius $R>0$,

$$\max_{y\in\widetilde{M}}|\widetilde{B}(y,R)|>V_{hyp}(R).$$

This is a macroscopic version of a conjecture of R. Schoen concerning scalar curvature. Guth proved the assertion for radius $R=1$ under the stronger hypothesis that the volume is sufficiently small compared with the hyperbolic volume; the conjecture asks for all positive radii under the stated volume hypothesis.