The convergence conjecture for intrinsic volumes under noncollapsed limits

Let $\{M_l^n\}_{l=1}^{\infty}$ be smooth $n$-dimensional closed connected Riemannian manifolds with a uniform lower bound on sectional curvature, converging in the Gromov–Hausdorff metric to a compact Alexandrov space $X$ without collapse. The noncollapsed convergence conjecture. For every $i=0,\dots,n$, the sequence $\{V_i(M_l)\}$ converges, and its limit depends only on $X$, not on the approximating sequence. The conjecture is known for $i=0$ by Perelman's stability theorem and was claimed for $i=n-2$ in work in progress; the remaining intrinsic volumes are generally open.