The integral scalar-curvature stability conjecture for Riemannian tori

Let $K\in\mathbb{R}$, $n\in\mathbb{N}$, and $v,D>0$. For every $\epsilon>0$, consider Riemannian $n$-tori $M$ satisfying $\operatorname{ric}_M\geq K$, $\operatorname{vol}_M(M)\geq v$, and $\operatorname{diam}_M\leq D$. Integral scalar-curvature stability conjecture. There exists $\delta>0$ such that, if additionally

$$\int_M R_M^-\,\operatorname{d}\!\operatorname{vol}_M\leq\delta,$$

then there exists an $n$-dimensional flat torus $T$ for which $M$ is $\epsilon$-Gromov-Hausdorff close to $T$ and diffeomorphic to $T$. The conjecture seeks a quantitative torus-stability result under a lower Ricci bound and a small integral negative scalar-curvature part.