Summable Hausdorff measure conjecture for singular strata of Alexandrov spaces

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, let $(X,p)\in\operatorname{Alex}^n(-1)$, and set

$$\epsilon_i=2^{-i}.$$

Here $\mathcal{S}^k_{\epsilon_i}$ denotes the $\epsilon_i$-singular stratum and $\mathcal{H}^k$ is the $k$-dimensional Hausdorff measure. Summable Hausdorff measure conjecture. One has

$$\sum_{i=0}^\infty \epsilon_{i+1}^{(n-k)-1}\mathcal{H}^{k}\Big(\big(\mathcal{S}^{k}_{\epsilon_{i+1}}\setminus\mathcal{S}^{k}_{\epsilon_i}\big)\cap B_1(p)\Big)<C(n).$$

This is presented as a stronger summable form of the preceding quantitative Hausdorff measure conjecture, with the source pointing to an example as motivation. Its status is not resolved in the provided source context.