Storm's characterization of geodesic boundary by local volume minimality

Let $M$ be a finite-volume hyperbolic $3$-manifold with minimal surface boundary, and interpret locally minimal among such manifolds with respect to volume.

Storm's local-minimality conjecture. If $M$ is locally minimal among such manifolds, then its boundary is totally geodesic.

The source proposes this as a possible route to reproving the main theorem by deforming minimal-surface boundary to geodesic boundary while decreasing volume; no resolution is supplied.