Almost rigidity of the Positive Mass Theorem for special surfaces

Fix $D>0$ and $r_0>0$. Let $\mathcal M$ be the class of asymptotically flat three-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces, with either no boundary or outermost minimizing boundary. Let $M_j^3\in\mathcal M$, let $\Sigma_j\subset M_j^3$ be special surfaces with $\operatorname{Area}(\Sigma_j)=4\pi r_0^2$, and let $\Sigma_\infty=\partial B(0,r_0)\subset\mathbb E^3$. Almost-rigidity conjecture. If

$$m_{ADM}(M_j)\to 0,$$

then

$$d_{\operatorname{Vol}\mathcal F}\bigl(T_D(\Sigma_j),T_D(\Sigma_\infty)\bigr)\to 0,$$

where $T_D(\Sigma)$ is the tubular neighborhood of radius $D$ around $\Sigma$; alternatively, if $\Omega_j$ is the interior of $\Sigma_j$ with $\operatorname{Depth}(\Omega_j)\le D$, then

$$d_{\operatorname{Vol}\mathcal F}(\Omega_j,\Omega_\infty)\to 0.$$

This is an intrinsic-flat formulation of almost rigidity for the Positive Mass Theorem and is related to Bartnik's conjecture; the general statement remains open.