The Q-curvature volume comparison conjecture for closed hyperbolic manifolds

Let $n\geq 3$, let $(M^n,\bar g)$ be a closed hyperbolic manifold, and let $g$ be a metric on $M$. Here $Q_g$ denotes the $Q$-curvature of $g$ and $V_M(g)$ its volume.

Q-curvature volume comparison conjecture. If

$$Q_g\geq Q_{\bar g},$$

then

$$V_M(g)\leq V_M(\bar g).$$

This proposes a higher-dimensional extension of the global volume comparison established in the paper for closed hyperbolic $4$-manifolds. The conjecture is presented as open for general dimensions $n\geq 3$.