Torus and Positive Mass stability for metrics with Ricci curvature lower bound

Consider a sequence of metrics $g_i$ on the torus whose members have uniform lower bounds on their first stable systoles and Ricci curvatures, and have a uniform upper bound on their diameters. If the $L^1$ norm of the negative part of the scalar curvatures vanishes along this sequence, then we show there is a subsequence of the metrics which converges in the measured-Gromov-Hausdorff topology to a flat metric on the torus. Something analogous holds for sequences of asymptotically flat spin Riemannian manifolds with a uniform lower bound on Ricci curvature and nonnegative scalar curvature: if the ADM masses of the distinguished ends tend to zero along the sequence, then the manifolds converge to Euclidean space in the pointed measured Gromov-Hausdorff sense.