Lee–Sormani intrinsic flat stability conjecture for asymptotically flat manifolds

Let $M_j$ be asymptotically flat $n$-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces, with either no boundary or boundary equal to an outermost minimizing surface. Fix $A_0>0$, and choose $p_jin\Sigma_j$ on a special surface $\Sigma_j\subset M_j$ satisfying

$$\operatorname{Vol}_{n-1}(\Sigma_j)=A_0.$$

If

$$\mathrm{m}_{\mathrm{ADM}}(M_j)\to 0,$$

then $(M_j,p_j)$ converges to Euclidean space $(\mathbb{E}^n,0)$ in the pointed intrinsic flat sense. That is, for almost every $D>0$,

$$d_{\mathcal{F}}\left(B_{p_j}(D)\subset M_j,B_0(D)\subset\mathbb{E}^n\right)\to 0.$$

This conjecture concerns intrinsic-flat stability of the rigidity case of the positive mass theorem. The stated version is false without careful selection of the points $p_j$: increasingly thin wells can contain the chosen points and prevent convergence to Euclidean space.