Penrose's Gibbons–Penrose inequality for surfaces in Minkowski spacetime

Let $\Sigma$ be a two-dimensional spacelike closed embedded orientable surface in Minkowski spacetime, diffeomorphic to $S^2$. Let $T_0\in\mathbb{R}^{3,1}$ be a fixed future timelike vector satisfying $\langle T_0,T_0\rangle=-1$. Let $\vec{H}$ be the mean curvature vector of $\Sigma$, and let $\vec{J}$ be its dual mean curvature vector. A surface is past null convex when the past null hypersurface generated by it is smooth. Penrose's conjecture. Suppose that $\Sigma$ is past null convex. Then

$$-\int_\Sigma \langle \vec{J},T_0\rangle\,d\mu\geq\sqrt{16\pi\,|\Sigma|}.$$

This is a spacetime analogue of the Penrose inequality, relating the dual mean-curvature integral to the area of the surface. The source does not state a resolution, so the conjecture is treated as open.