The positive mass conjecture for asymptotically hyperbolic manifolds with boundary

Let $(M,g,\Sigma)$ be an asymptotically hyperbolic manifold with scalar curvature $R_g$ and boundary mean curvature $H_g$ satisfying

$$R_g\geq -n(n-1),\qquad H_g\geq 0.$$

For any admissible chart $F$, let $\mathcal P^{[F]}$ be the mass vector, regarded in the Lorentzian space associated with the isometry group of $(\mathbb H^n_+,b,\partial\mathbb H^n_+)$. Positive mass conjecture. For every admissible chart $F$, the vector $\mathcal P^{[F]}$ is time-like and future directed, unless it vanishes and $(M,g,\Sigma)$ is isometric to $(\mathbb H^n_+,b,\partial\mathbb H^n_+)$.

This conjecture is the asymptotically hyperbolic analogue of the positive mass statement, with the hyperbolic half-space as the reference geometry and rigidity at zero mass. The supplied text does not state whether the conjecture has been resolved.