The Penrose inequality for asymptotically flat initial data

Let $(M,g,k)$ be an asymptotically flat initial data set satisfying appropriate fall-off conditions and the dominant energy condition, with boundary $\partial M$ consisting of an outermost apparent horizon, possibly with multiple components. Let $N$ be any component of the boundary, and let $A=A_{\min}(N)$ denote the outermost minimal area enclosure of $N$. Penrose inequality. The ADM energy satisfies

$$E_{ADM}\geq \sqrt{\frac{A}{16\pi}}.$$

Here $E_{ADM}$ is the ADM energy. The Riemannian case is known, as are certain special cases such as spherical symmetry, but the inequality for general asymptotically flat initial data with extrinsic curvature remains open.