Almost rigidity of the Scalar Torus Theorem

Let $M_j^3$ be diffeomorphic to $\mathbb T^3$ and satisfy

$$\operatorname{Scalar}(M_j^3)\ge -1/j,$$

and

$$\operatorname{Vol}(M_j^3)=V_0,\qquad \operatorname{Diam}(M_j^3)\le D_0,\qquad \operatorname{MinA}(M_j^3)\ge A_0>0.$$

Almost-rigidity conjecture. A subsequence should converge intrinsically flatly to a flat torus:

$$M_{j_k}\stackrel{\mathcal F}{\longrightarrow}M_0,$$

and possibly even converge with volume convergence. The conjecture is motivated by the Scalar Torus Theorem; it is false for Gromov–Hausdorff convergence and remains open for intrinsic-flat convergence.