The intrinsic volumes conjecture for weakly smoothable Alexandrov spaces

Let $X$ be a compact weakly smoothable Alexandrov space, meaning that it is a Gromov–Hausdorff limit of closed smooth connected Riemannian manifolds with uniformly bounded above dimension and a uniform lower bound on sectional curvature. For each stratum $S$ of the extremal stratification, seek numbers $V_i(S)$, $i=0,1,2,\dots$, called intrinsic volumes. The intrinsic volumes conjecture. These numbers should satisfy: $V_i(S)=0$ for $i>\dim_H S$; $V_{\dim_H S}(S)$ equals the $\dim_H S$-dimensional Hausdorff measure of $S$; and

$$V_0(S)=\chi_c(S)=\sum_k(-1)^k\dim H_c^k(S,\mathbb{F}),$$

where $\mathbb{F}$ is any field. Moreover, for a sequence $M_l^n$ with a uniform lower curvature bound converging to $X$, after choosing the subsequence supplied by the constructible-function theorem and writing $F=\sum_k(-1)^kF_k$, one should have

$$\lim_{l\longrightarrow\infty}V_i(M_l)=\sum_S F(S)\,V_i(S),$$

where the sum is over the strata of the extremal stratification and $F(S)$ is the value of $F$ on $S$. This conjecture refines the noncollapsed and collapsing convergence conjectures and would define intrinsic volumes for a broad class of singular Alexandrov spaces; finiteness of the relevant compactly supported cohomology groups is itself only expected.