Large-volume capacity minimizer conjecture

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass, and assume that $\partial M$ is empty or minimal. For $V>0$ sufficiently large, consider compact regions containing $\partial M$ and having volume $V$. Large-volume capacity minimizer conjecture. There exists a unique such compact region whose capacity is minimized among all such regions.

This is the capacity analogue of the existence and uniqueness of large isoperimetric regions in asymptotically flat 3-manifolds. The conjecture remains open according to the supplied source context.