Stability of the positive mass theorem for special-surface tubular neighborhoods

Let $\mathcal{M}$ be a subclass of asymptotically flat three-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces, with either no boundary or boundary an outermost minimizing surface. For $\alpha_0>0$, let $\Sigma_{\alpha_0}$ be a special surface with

$$\operatorname{Vol}_{2}(\Sigma_{\alpha_0})=\alpha_0,$$

and let $T_D(\Sigma_{\alpha_0})$ denote its tubular neighborhood of radius $D$. Let $\mathbb{E}^3$ be Euclidean space. Stability conjecture. Given any $\epsilon>0$, $D>0$, and $\alpha_0>0$, there exists $\delta=\delta(\epsilon,D,\alpha_0)>0$ such that if $M^3\in\mathcal{M}$ has ADM mass $\operatorname{m}_{\mathrm{ADM}}(M)<\delta$, then

$$d_{\mathcal{F}}\bigl(T_D(\Sigma_{\alpha_0})\subset M^3,\,T_D(\Sigma_{\alpha_0})\subset\mathbb{E}^3\bigr)<\epsilon.$$

This proposes intrinsic flat stability of the positive mass theorem in the stated three-dimensional class, measured on fixed-radius tubular neighborhoods of special surfaces. The supplied text does not indicate whether the conjecture has been resolved.