Cheeger--Colding volume-ratio conjecture for collapsed Ricci limit spaces

Let $(X,\mathsf{d})$ be the metric limit under consideration, let $k=\dim_{\mathcal{H}}(X,\mathsf{d})$, and let $\mathcal{H}^k$ denote its $k$-dimensional Hausdorff measure. For $m\in[k,n]\cap\mathbb{N}$, let $\mathrm{Vol}_m(r)$ be the volume of a radius-$r$ ball in the $m$-dimensional space form of sectional curvature $-1$. Cheeger--Colding conjecture. There exists $m\in[k,n]\cap\mathbb{N}$ such that, for every $y\in X$,

$$\frac{\mathcal{H}^k(B_r(y))}{\mathrm{Vol}_m(r)}\downarrow\qquad(r\uparrow\infty).$$

This is presented as a conjecture raised by Cheeger and Colding concerning asymptotic volume ratios; the source gives no resolution.