Intrinsic flat almost rigidity of the hyperbolic positive mass theorem

Let $M_j^m$ be a sequence of asymptotically hyperbolic Riemannian manifolds with scalar curvature $R\geq -m(m-1)$ and mass tending to zero. Assume that $M_j^m$ contains no closed interior minimal surfaces, and that either $M_j^m$ has no boundary or its boundary is a closed minimal surface. Fix $A_0>0$ and $D>0$, and let $\Sigma_j\subset M_j$ be constant mean curvature surfaces of fixed area $A_0$. For a hypersurface $\Sigma$, write $T_D(\Sigma)$ for its tubular neighborhood of radius $D$, and let $\Sigma_\infty$ be a geodesic sphere in hyperbolic space $\mathbb{H}^m$ with $\operatorname{Vol}_{m-1}(\Sigma_\infty)=A_0$, centered at a point $p_\infty$. Intrinsic flat almost rigidity of the hyperbolic positive mass theorem. The tubular neighborhoods satisfy

$$\lim_{j\to\infty}d_{\mathcal{F}}\left(T_D(\Sigma_j)\subset M_j^m,T_D(\Sigma_\infty)\subset\mathbb{H}^m\right)=0$$

and

$$\lim_{j\to\infty}\operatorname{Vol}\left(T_D(\Sigma_j)\subset M_j^m\right)=\operatorname{Vol}\left(T_D(\Sigma_\infty)\subset\mathbb{H}^m\right).$$

Consequently, if $p_j\in\Sigma_j$, then $(M_j,p_j)$ converges in the pointed intrinsic flat sense to $(\mathbb{H}^m,p_\infty)$. This conjecture is an intrinsic-flat stability statement for the positive mass theorem in the asymptotically hyperbolic setting: vanishing mass should force convergence to hyperbolic space once points are anchored on fixed-area CMC surfaces. The parser supplies no evidence that it has been resolved, so its status remains open.