Sormani's intrinsic-flat stability conjecture for almost nonnegative scalar curvature

Let $M_j$ be a sequence of Riemannian manifolds diffeomorphic to a $3$-torus such that

$$\operatorname{vol}(M_j)=V_0,\qquad \operatorname{diam}(M_j)\leq D_0,\qquad \operatorname{minA}(M_j)\geq A_0>0,$$

where

$$\operatorname{minA}(M_j)=\inf\{\operatorname{vol}(\Sigma):\Sigma\text{ is a closed minimal surface in }M_j\}.$$

If the scalar curvature $R(M_j)$ satisfies $R(M_j)\geq -\frac{1}{j}$, then Sormani's conjecture. There is a subsequence $M_{j_k}$ converging in intrinsic flat sense to a flat torus $T$, and possibly $\operatorname{vol}(M_{j_k})\to\operatorname{vol}(T)$. The conjecture refines scalar-curvature stability by imposing volume, diameter, and minimal-area noncollapsing bounds; the supplied text does not state that it has been resolved.