The all-radii hyperbolic volume comparison conjecture

Let $(M^n,hyp)$ be a closed hyperbolic manifold, and let $g$ be another metric on $M$ with $Vol(M,g)<Vol(M,hyp)$. Hyperbolic volume comparison conjecture.

$$V_{(\widetilde M,\widetilde g)}(R)>V_{\mathbb H^n}(R)$$

for all $R$. This strengthens the asymptotic comparison theorem discussed in the paper by requiring the inequality at every radius; the supplied text gives no resolution status.