Li–Naber integral singularity conjecture for Alexandrov spaces

Let $X\in\text{Alex}^n(-1)$. For $0\le k\le n-1$, define the $k$-singularity function by setting $\theta_k(x)=2\pi$ for $x\in\mathcal{S}^{k-1}(X)$ and $\theta_k(x)=2\pi-\operatorname{diam}(\Sigma^{n-k-1}_x)$ when $x\in\mathcal{S}^k(X)\setminus\mathcal{S}^{k-1}(X)$ has tangent cone $T_x(X)=\mathds{R}^k\times C(\Sigma^{n-k-1}_x)$. Li–Naber's integral singularity conjecture. One has

$$\int_{B_1}\theta_k^{n-k-1}(x)\cdot \operatorname d\mathcal{H}^k<C(n).$$

In particular, for $k=n-2$,

$$\int_{B_1}\theta_{n-2}(x)\cdot \operatorname d\mathcal{H}^{n-2}<C(n).$$

This is a summable strengthening of the preceding quantitative stratification estimate. The source states that the $k=n-2$ case is known for non-collapsed limits of manifolds with lower sectional-curvature bounds, while the general conjecture remains open.