Ilmanen's quantitative stability conjecture for the positive mass theorem

Let $(M_i,g_i)$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass $m(g_i)$ tending to $0$. Suppose $Z_i\subset M_i$ are the subsets whose removal yields Gromov–Hausdorff convergence to Euclidean $3$-space as in the Huisken–Ilmanen stability conjecture. Ilmanen's quantitative stability conjecture. The subsets $Z_i$ can be chosen so that

$$\operatorname{Area}(\partial Z_i)\leq 16\pi m(g_i)^2.$$

The proposed estimate is motivated by the Penrose inequality and strengthens qualitative stability by prescribing an explicit upper bound for the exceptional boundary area in terms of the ADM mass. Its resolution is not supplied in the source.