Weak convergence of curvature tensors under Gromov–Hausdorff convergence

Let $\mathcal M(n,r,V)$ be the class of $n$-dimensional complete Riemannian manifolds with sectional curvature at most $1$, injectivity radius at least $r>0$, and volume at most $V$. Suppose that $M_j\in\mathcal M(n,r,V)$ converge to a metric space $M$ in the Gromov–Hausdorff topology. Weak convergence conjecture. The curvature tensors of $M_j$ weakly converge as measures. In particular, the total scalar curvatures of $M_j$ converge. If $M$ is a Riemannian manifold, then the limit is the total scalar curvature of $M$. This is the expected analogue, for upper sectional-curvature bounds, of the convergence theorem of Lebedeva and Petrunin for manifolds with sectional curvature bounded below and without collapsing; the statement concerns curvature measures and, in particular, convergence of total scalar curvature.