Naber's curvature-measure convergence conjecture

Let $(M_i,g_i)$ be a sequence of $n$-dimensional manifolds with lower Ricci or sectional curvature bounds, and suppose that $M_i$ Gromov–Hausdorff converge without collapsing to $X$. Naber's curvature-measure convergence conjecture. As measures,

$$scal_i \,\operatorname d {vol}_{g_i}\to {\operatorname R}\,\operatorname d\mathcal{H}^n+\Phi\,\operatorname d\mathcal{H}^{n-1}+\theta\,\operatorname d\mathcal{H}^{n-2},$$

where ${\operatorname R}$, $\Phi$, and $\theta$ are locally $L^1$ with respect to $\mathcal{H}^n$, $\mathcal{H}^{n-1}$, and $\mathcal{H}^{n-2}$, respectively; $\Phi$ is supported on an $(n-1)$-rectifiable subset; and $\theta(x)=\theta_{n-2}(x)$ is supported on the top stratum of the $(n-2)$-rectifiable singular set $\mathcal{S}(X)=\mathcal{S}^{n-2}(X)$. The conjecture seeks a curvature-measure decomposition for nonsmooth Gromov–Hausdorff limits. The source notes partial results for lower sectional-curvature bounds, but leaves the general formulation open.