Growth in the universal cover under large simplicial volume

Consider a closed manifold $M$ with two Riemannian metrics: one hyperbolic metric, and one other metric $g$. What hypotheses on $g$ guarantee that for a given radius $r$, there are balls of radius $r$ in the universal cover of $(M,g)$ with greather-than-hyperbolic volumes? We show that this conclusion holds for all $r\ge 1$ if $(\mathrm{V}\mathrm{o}\mathrm{l}(M,g){)}^2$ is less than a small constant times the hyperbolic volume of $M$. This strengthens a theorem of Sabourau and is partial progress toward a conjecture of Guth.