Bray and Khuri's generalized Penrose inequality

Let $(\Sigma,\gamma_{ij},A_{ij})$ be an asymptotically Euclidean initial data set that is Schwarzschild at infinity, with total mass $M_{ADM}$ in a chosen end, and satisfying the dominant energy condition $\rho\geq |\vec{J}|$. Let $S$ be a closed surface bounding an open set $\Omega$ containing all asymptotically Euclidean ends except the chosen one. Assume that $S$ is a generalized trapped surface with respect to the normal pointing towards the chosen end. Let $S_{\min}=\partial\Omega_{\min}$ be the minimal-area enclosure of $S$, meaning that $\Omega\subset\Omega_{\min}$ and $S_{\min}$ has least area among all surfaces with this property. Bray and Khuri's conjecture. One should have

$$M_{ADM}\geq\sqrt{\frac{|S_{\min}|}{16\pi}}.$$

Equality occurs if and only if $(\Sigma\setminus\Omega_{\min},\gamma_{ij},A_{ij})$ is the induced data of an embedding of $\Sigma\setminus\Omega_{\min}$ into the Kruskal spacetime such that $S_{\min}$ is mapped to a generalized apparent horizon. This is a proposed version of the Penrose inequality for general initial data; its status is not established in the supplied text.