Generalized nonnegative scalar curvature under intrinsic-flat convergence

Suppose a sequence of three-dimensional manifolds satisfies

$$M_j^3\stackrel{\mathcal F}{\longrightarrow}M_\infty \quad\text{or}\quad M_j^3\stackrel{V\mathcal F}{\longrightarrow}M_\infty,$$

with $\operatorname{Scalar}_j\ge 0$ and

$$\operatorname{MinA}(M_j)\ge A_0>0,\qquad \operatorname{Vol}(M_j)\in[V_0,V_1]\subset(0,\infty),\qquad \operatorname{Diam}(M_j)\le D_0.$$

Generalized scalar-curvature conjecture. Then: (a) $M_\infty$ is a nonzero integral current space, with no cancellation without collapse; (b) $M_\infty$ is geodesic; (c) angles between geodesics emanating from a point exist in a suitable sense; (d) dihedral angles between intersecting surfaces exist in a suitable sense; (e) Gromov's Gauss–Bonnet Prism Inequality holds on $M_\infty$; and (f) for every $p\in M_\infty$ there exists $r_p>0$ such that for every $r<r_p$,

$$V_p(r)=\frac{{\mathbf M}(B(p,r))}{4\pi r^3/3}\le 1.$$

These properties would provide a generalized notion of nonnegative scalar curvature on the limit space. The conjecture is open.