Gromov's Sobolev-type stability conjecture for almost nonnegative scalar curvature

Let $X$ be an $n$-manifold with Riemannian metric $g$, and let $R_g$ denote its scalar curvature. A Sobolev-type weak metric is a weak metric on the space of $n$-manifolds. Gromov's conjecture. There is a particular Sobolev-type weak metric such that, for example, tori $(X,g)$ with $R_g\geq -\epsilon$, when properly normalized, (sub)converge to flat tori as $\epsilon\to 0$, although these $X$ may in general diverge in stronger metrics. This conjecture formulates a stability problem for the rigidity of tori with nonnegative scalar curvature; the supplied text gives no resolution, so its status remains open.