Volume bound for guts of hyperbolic 3-manifolds

Let $M$ be a hyperbolic manifold containing an incompressible surface $S$. For a manifold $N$ with acylindrical boundary, let $Vol(N)$ denote the volume of the components of $N$ that admit a hyperbolic metric with totally geodesic boundary. Write $Guts(M\setminus\mathcal{N}(S))$ for the pared acylindrical manifold pair obtained from the complement of a regular neighborhood of $S$. Volume conjecture.

$$Vol(M)\geq Vol(Guts(M\setminus\mathcal{N}(S))).$$

This would sharpen the previously established lower bound in terms of the Euler characteristic of the guts, replacing it with the hyperbolic volume of the guts themselves. The supplied text gives no evidence that the conjecture has been resolved.