Uniform Comparison of Hyperbolic Ball Volumes on the Universal Cover

Let $\|M\|_{\Delta}$ denote the simplicial volume of $M$, $V_r(X,h)=\sup_{x\in X}\operatorname{Vol}_h\big(B_h(x,r)\big)$, and $\mathbb{H}^n$ denotes hyperbolic $n$-space. We prove that, if a closed oriented $n$-manifold $M$ admits a hyperbolic metric, then there is a dimensional constant $\delta_n>0$ such that every Riemannian metric $g$ on $M$ with $$\frac{\operatorname{Vol}_g(M)}{\|M\|_{\Delta}}<\delta_n$$ satisfies $$V_r(\widetilde M,\widetilde g)\ge V_r(\mathbb{H}^n) \quad\text{for every }r\ge 1.$$