Li–Naber quantitative stratification conjecture for Alexandrov spaces

Let $\text{Alex}^n(-1)$ denote the $n$-dimensional Alexandrov spaces with curvature at least $-1$. For $0\le k\le n-1$, let $\mathcal{S}^k_\epsilon(X)$ be the set of points whose tangent cone is $\epsilon$-away from splitting off $\mathds{R}^{k+1}$, and let $B_1$ be the unit ball. Li–Naber's conjecture. There is a constant $C(n)$ such that, for every $X\in\text{Alex}^n(-1)$,

$$\mathcal{H}^{k}(\mathcal{S}^{k}_\epsilon(X)\cap B_1)<C(n)\epsilon^{1-(n-k)}.$$

This would sharpen the known bound $\mathcal{H}^k(\mathcal{S}^k_\epsilon(X)\cap B_1)\le C(n,\epsilon)$ by specifying its dependence on $\epsilon$. The source attributes the conjecture to Li and Naber and gives no resolution.