Maldacena's negative-volume Schottky filling conjecture

A Riemann surface is a connected one-dimensional complex manifold. A Schottky manifold is a convex co-compact hyperbolic $3$-manifold whose conformal boundary is a Riemann surface. Its renormalized volume is the regularized hyperbolic volume associated with the asymptotic boundary.

Maldacena's conjecture. Every connected Riemann surface of genus at least $2$ is the asymptotic boundary of a Schottky manifold with negative renormalized volume.

The conjecture arises from the AdS/CFT correspondence, where the dominant bulk contribution is expected to come from a filling with smallest renormalized volume. It asserts that every such surface admits a Schottky filling whose renormalized volume is negative; the supplied source does not state whether this has been proved or disproved.