Positive energy-momentum conjecture with rigidity for asymptotically hyperbolic manifolds

Let $(M,g)$ be a complete, $n$-dimensional Riemannian asymptotically hyperbolic manifold whose conformal infinity is the $(n-1)$-unit sphere. Let $R_g$ denote the scalar curvature of $g$, and let $\boldsymbol{p_g}$ denote its energy-momentum vector. Positive energy-momentum conjecture. If

$$R_g\geq -n(n-1),$$

then $\boldsymbol{p_g}$ is timelike and future-directed, unless $(M,g)$ is isometric to hyperbolic space. This is the general positive energy-momentum statement in the asymptotically hyperbolic setting, with equality characterized by the hyperbolic space; the paper states that this general result remains an open question.