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This question is a natural follow up to the question asked here. I think it should not be too hard to answer it (negatively) in dimension 3, but higher dimensions will be probably challenging.

Let $M$ be a connected hyperbolic $n$-manifold. A round ball in $M$ is an open metric $R$-ball $B=B(p,R)\subset M$ which admits a Riemannian isometry to an open $R$-ball in the hyperbolic space ${\mathbb H}^n$. In other words, the exponential map $\exp_p: T_pM\to M$ is a diffeomorphism from $B(0,R)\subset T_pM$ to $B$.

Question. Is there a function $V(n,m)$ such that for every compact hyperbolic $n$-manifold $M$, $n\ge 3$, covered by $m$ round metric balls, the volume of $M$ is $\le V(n,m)$?

Note that the answer is negative in the case of hyperbolic surfaces (even $m=3$ suffices), see here.

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