De Philippis--Gigli characterization of non-collapsed RCD spaces

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ space, and let $\dim(X,\mathsf{d},\mathfrak{m})$ be its essential dimension. De Philippis--Gigli conjecture. If

$$\dim(X,\mathsf{d},\mathfrak{m})=N,$$

then $\mathfrak{m}=a\mathcal{H}^N$ for some $a\in(0,\infty)$. This would characterize, up to a positive rescaling of the reference measure, the non-collapsed $\operatorname{RCD}(K,N)$ spaces; it is known when $(X,\mathsf{d})$ is compact, but remains open in general.