RCD compactness and homeomorphism conjecture for singular Kähler spaces

Let $X$ be an $n$-dimensional projective variety with log terminal singularities, and let $\theta_X$ be a smooth Kähler metric on $X$. Let $(X,\omega)\in\mathcal{RK}(n)$, so that $obreak\omega\in\mathcal{RK}(X)$, and let $(\hat X,d_\omega,\mu_\omega)$ be the metric measure space induced by $(X,\omega)$ as in Definition. RCD conjecture. For any $(X,\omega)\in\mathcal{RK}(n)$, the metric measure space $(\hat X,d_\omega,\mu_\omega)$ is a compact $\operatorname{RCD}$ space homeomorphic to the projective variety $X$ itself. The conjecture proposes that the pluripotential-theoretic Ricci lower bound defining $\mathcal{RK}(X)$ yields an RCD structure and that the metric completion retains the topology of the underlying projective variety; the statement is presented as a conjecture, with no resolution given here.