Sormani's scalar curvature conjecture for intrinsic flat limits

Let $M_j^3$ be a sequence of three-dimensional Riemannian manifolds. Write ${\rm Scal}_j$ for the scalar curvature, $\operatorname{Vol}(M_j)$ for their volume, and $\textrm{MinA}(M_j)$ for the minimal area of a closed minimal surface in $M_j$. Assume

$${\rm Scal}_j\ge 0,\qquad \operatorname{Vol}(M_j)\ge V_0>0,\qquad \textrm{MinA}(M_j)\ge A_0>0.$$

Suppose that $M_j$ converges in the intrinsic flat sense to $M_\infty$, written $M_j\stackrel{\mathcal{F}}{\longrightarrow}M_\infty$. For $p\in M_\infty$, let $B(p,r)$ be the metric ball in the limit space, let $\operatorname{Vol}_{Y}$ denote its volume, and let $\operatorname{Vol}_{\mathbb{E}^3}(B(0,r))$ denote the Euclidean volume of a radius-$r$ ball. Sormani's scalar curvature conjecture. At every point $p\in M_\infty$,

$$\lim_{r\to 0}\frac{\operatorname{Vol}_{\mathbb{E}^3}(B(0,r))-\operatorname{Vol}_{Y}(B(p,r))}{r^2\operatorname{Vol}_{\mathbb{E}^3}(B(0,r))}\ge 0.$$

This is proposed as a possible revision of Gromov's vague conjecture, motivated by counterexamples showing that intrinsic flat and Gromov--Hausdorff limits of noncollapsing sequences with positive scalar curvature may fail to retain key properties of nonnegative scalar curvature. The statement remains open in the supplied source.