Huisken's Penrose inequality conjecture for asymptotically flat support surfaces

Let $S$ be an asymptotically flat support surface with mean curvature $H^S\geq 0$. Let $S'\subset S$ be an exterior surface with free boundary minimal surface $\Sigma$ and exterior mass $m_{ext}$; assume that $\Sigma$ is connected. Huisken's conjecture. One should have

$$m_{ext}\geq\sqrt{\frac{|\Sigma|}{\pi}}$$

with equality if and only if $S'$ is a half-catenoid and $\Sigma$ is the free boundary disc contained in the symmetry plane of the catenoid. This is the precise formulation attributed to Volkmann's thesis and gives the support-surface analogue of the Riemannian Penrose inequality; its resolution is not stated in the supplied text.