RCD homeomorphism conjecture for singular Kähler varieties

Let $X$ be a normal Kähler variety with klt singularities. Let $\omega$ be a Kähler current whose volume measure has an $L^p$ density for some $p>1$, whose Ricci current is bounded below by $-\omega$, and which is smooth on a Zariski open subset $X^\circ$ of $X$. Consider the metric measure space $(X^\circ,\omega,\omega^n)$.

RCD homeomorphism conjecture. The metric completion of $(X^\circ,\omega,\omega^n)$ is a non-collapsed RCD space homeomorphic to $X$.

This conjecture would connect Kähler geometry on singular varieties with the topology and metric structure of RCD spaces. The source introduces it as a main question in the developing theory, with no resolution stated.