Gromov's conjecture on convergence of almost nonnegatively scalar-flat tori

Let $M_j=(\mathbb{T}^n,g_j)$, with $n\geq 3$, be a sequence of Riemannian manifolds, where $R_{g_j}$ denotes the scalar curvature. Assume

$$R_{g_j}\geq -\frac{1}{j},\qquad V_0\leq \operatorname{Vol}(M_j)\leq V^0,\qquad \operatorname{Diam}(M_j)\leq D_0.$$

Gromov's conjecture. If no bubbling occurs along the sequence, or if the bubbles forming along the sequence can be cut out, then a subsequence of $M_j$ converges in some sense to the flat torus.

The conjecture concerns compactness and convergence for sequences of metrics on tori whose scalar curvature approaches nonnegativity while volume and diameter remain uniformly controlled. The precise meaning of convergence and the treatment of bubbling are the issues left open in the stated formulation.