The RCD conjecture for singular Kähler metric completions

Let $(X,\omega_X)$ be an $n$-dimensional compact normal Kähler space with smooth Kähler metric $\omega_X$, and let $\omega=\omega_X+\sqrt{-1}\partial\bar\partial u$ be a singular Kähler metric. Assume that $u$ is smooth on $X^\circ=X^{reg}\setminus D$ for a divisor $D$, that $\omega^n=e^F\omega_X^n$ with $F\in L^1(\omega_X^n)$ and $e^F\in L^p(\omega_X^n)$ for some $p>1$, and that $\operatorname{Ric}(\omega)>-A(\omega+\omega_X)$ for some $A>0$ on $X^{reg}$. Let $\hat X$ be the metric completion of $(X^\circ,\omega)$. RCD conjecture. If $(X,\omega)$ satisfies conditions (1), (2), and (3), then the metric completion $\hat X$, equipped with the measure $\omega^n$, is an RCD space. The theorem in the paper proves the corresponding homeomorphism and Hausdorff-dimension conclusions under an RCD assumption; the conjecture asserts that this assumption is superfluous, extending earlier conjectures for singular Kähler-Einstein metrics.