The positive mass conjecture with a noncompact boundary

Let $N^n$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ and order $p>\frac{n-2}{2}$, with integrable scalar curvature $R_g$ and integrable mean curvature $H_g$ on its noncompact boundary. Suppose that an end of $N$ is modeled on $\mathbb{R}^n_+$ outside a compact set, and let $m(g)$ be the corresponding mass. Positive mass conjecture with a noncompact boundary. If $R_g,H_g\geq 0$, then

$$m(g)\geq 0,$$

and equality holds if and only if $N$ is isometric to $\mathbb{R}^n_+$. The source notes that the corresponding theorem is available in the spin case through the cited boundary extension, but does not state a complete resolution of the conjecture in general.