De Philippis–Gigli conjecture on weakly non-collapsed RCD spaces

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ space, meaning that it satisfies the Riemannian curvature-dimension condition with parameters $K$ and $N$. It is weakly non-collapsed if $\mathfrak{m}\ll\mathcal{H}^N$, where $\mathcal{H}^N$ denotes the $N$-dimensional Hausdorff measure.

De Philippis–Gigli conjecture. If $(X,\mathsf{d},\mathfrak{m})$ is a weakly non-collapsed $\operatorname{RCD}(K,N)$ space, then

$$\mathfrak{m}=a\mathcal{H}^N$$

for some $a\in(0,\infty)$.

The conjecture asserts that weak non-collapsedness is equivalent, up to a constant normalization of the measure, to non-collapsedness. The source gives no evidence of a resolution.