Finite-volume scalar-curvature volume inequality

Let $(M,g)$ be a complete finite-volume $3$-manifold with minimal surface boundary, scalar curvature $R(g)\geq -6$, and let $DM$ denote its double. Write $\|DM\|$ for the simplicial volume of $DM$.

Finite-volume volume conjecture. The volume satisfies

$$\operatorname{Vol}(M,g)\geq \frac{1}{2}\|DM\|,$$

with equality if and only if $M$ has geodesic boundary.

The source identifies this as an unresolved case needed to extend the main theorem to noncompact manifolds; it specifically notes difficulty proving the theorem in this setting.