Huisken–Ilmanen Gromov–Hausdorff stability conjecture for the positive mass theorem

Let $M_i$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass tending to zero. Huisken–Ilmanen's conjecture. There is a set $Z_i e M_i$ such that $|\partial Z_i|\to 0$ and $M_i\setminus Z_i$ converges to $\mathbb{R}^3$ in the Gromov–Hausdorff topology. This conjecture asks for a stability version of the three-dimensional positive mass theorem in Gromov–Hausdorff topology; the source states that the appropriate topology for this problem was not known.