Gibbons–Penrose inequality for surfaces in Schwarzschild spacetime

Let $\Sigma$ be a spacelike $2$-surface in Schwarzschild spacetime, and let $\vec{J}$ denote its dual mean curvature vector. Suppose that the past null hypersurface generated by $\Sigma$ is smooth. Let $m$ be the total mass of the Schwarzschild spacetime. Gibbons–Penrose conjecture. Then

$$-\int_\Sigma \left\langle \vec{J},\frac{\partial}{\partial t}\right\rangle\,d\mu+16\pi m\geq\sqrt{16\pi|\Sigma|}.$$

This extends the Minkowski-space inequality by including the Schwarzschild mass term. The source does not state a resolution, so the conjecture is treated as open.