Quantitative Hausdorff measure conjecture for singular strata of Alexandrov spaces

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, and let $(X,p)\in\operatorname{Alex}^n(-1)$, where $\mathcal{S}^k_\epsilon$ denotes the $\epsilon$-singular stratum. Hausdorff measure conjecture. There is a constant $C(n)>0$ such that

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

The conjecture proposes an explicit dependence on $\epsilon$ for the Hausdorff measure estimate supplied by the preceding corollary. Its status is not resolved in the provided source context.