Belolipetsky–Emery conjecture on compact and noncompact hyperbolic manifolds

Let $cmathcal{M}$ be a compact hyperbolic manifold of dimension $n eq 3$. Belolipetsky–Emery conjecture. There \exists a noncompact hyperbolic $n$-manifold $cmathcal{M}_1$ whose volume is smaller than the volume of $cmathcal{M}$. The conjecture is motivated by the special role of dimension $3$, where hyperbolic Dehn filling can produce compact hyperbolic manifolds of smaller volume; it asserts the analogous comparison in every dimension other than $3$.