Almaraz–Barbosa–de Lima positive mass conjecture with non-compact boundary

Let $(M,g)$ be an asymptotically flat Riemannian manifold with non-compact boundary, scalar curvature $R$ and boundary mean curvature $H$. Almaraz–Barbosa–de Lima conjecture. If $(M, g)$ is asymptotically flat and satisfies $R \ge 0$ and $H \ge 0$, then

$$m(M,g) \ge 0.$$

Equality occurs if and only if $(M, g)$ is isometric to $(\mathbb{R}^n_+, \delta)$. The conjecture is a positive mass theorem for asymptotically flat manifolds with non-compact boundary; the source notes that it was proved when either $3 \le n \le 7$ or $n \ge 3$ and $M$ is spin, leaving the general case open.