Guth's normalized volume conjecture for uniform hyperbolic ball comparison

For a closed oriented connected $n$-dimensional manifold $M$ admitting a hyperbolic metric, let $g$ be a Riemannian metric on $M$, let $\|M\|_{\Delta}$ denote its Gromov simplicial volume, and write $(\widetilde M,\widetilde g)$ for the universal cover with the lifted metric. Let $V_r(X,h)$ be the supremum of the $h$-volumes of radius-$r$ balls in $(X,h)$. Guth's normalized volume conjecture. For every $n\ge2$, there exists $\delta_n>0$ such that, if

$$\frac{\operatorname{Vol}_g(M)}{\|M\|_{\Delta}}<\delta_n,$$

then

$$V_r(\widetilde M,\widetilde g)\ge V_r(\mathbb{H}^n)$$

for every $r\ge1$. This is an intermediate scale-uniform statement combining the covering-invariant hypothesis of Guth's fixed-scale theorem with the all-scale conclusion of Sabourau's theorem; its general validity remains open.