The RCD conjecture for singular Kähler–Einstein metrics

Let $X$ be an $n$-dimensional normal compact Kähler space, and let $\omega_{KE}$ be a singular Kähler–Einstein metric on $X$ satisfying

$$\operatorname{Ric}(\omega_{KE})=\lambda\omega_{KE}$$

on the regular set $X^{reg}$, with induced length metric $d_{KE}$ and metric completion $\overline{(X^{reg},d_{KE})}$. Equip this completion with the measure $\omega_{KE}^n$, extended trivially from $X^{reg}$. RCD conjecture. The metric space $\overline{(X^{reg},d_{KE})}$ is a non-collapsed $RCD(\lambda,2n)$-space and is homeomorphic to $X$. This conjecture proposes that singular Kähler–Einstein metrics satisfy the same kind of synthetic Ricci-curvature and dimension bounds as Riemannian manifolds with Ricci curvature bounded below. The statement is presented as likely folklore, but the source notes that it was not found in the literature in this generality; its resolution status is therefore open.