Geometric stability of the Schoen–Yau zero mass rigidity theorem

Let $M_j^3$ be a sequence of three-dimensional manifolds in the class $\mathcal{M}$, and let $m_{ADM}(M_j^3)$ denote their ADM masses. Let ${\mathbb E}^3$ denote Euclidean three-space. Geometric stability of the zero mass rigidity theorem. If

$$\operatorname{Scal}\ge 0 \quad\text{and}\quad m_{ADM}(M_j^3)\to 0,$$

then $M_j^3$ should converge to ${\mathbb E}^3$ with respect to some geometric notion of convergence. This is the proposed stability analogue of the Schoen–Yau zero mass rigidity theorem; the source leaves the precise geometric convergence notion unspecified, and the conjecture remains open.