Growth in the universal cover under large simplicial volume

Consider a closed manifold [Formula: see text] with two Riemannian metrics: One hyperbolic metric, and one other metric [Formula: see text]. What hypotheses on [Formula: see text] guarantee that for a given radius [Formula: see text], there are balls of radius [Formula: see text] in the universal cover of [Formula: see text] with greater-than-hyperbolic volumes? We show that this conclusion holds for all [Formula: see text] if [Formula: see text] is less than a small constant times the hyperbolic volume of [Formula: see text]. This strengthens a theorem of Sabourau and is partial progress toward a conjecture of Guth.