Homeomorphism conjecture for the generalized Kähler–Einstein metric space

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample canonical line bundle $K_X$. For sufficiently large $\ell$, let $\pi:X\rightarrow X_{can}\subset\mathbb{C}P^N$ be the holomorphic Calabi–Yau fibration onto its canonical model, and let $S$ be the set of singular values. Let $\omega_{\operatorname{GKE}}$ be the generalized Kähler–Einstein current, let $d_{\operatorname{GKE}}$ be its induced length metric on $X_{can}\backslash S$, and define $X_{\operatorname{GKE}}$ as the metric completion of $(X_{can}\backslash S,d_{\operatorname{GKE}})$. Homeomorphism conjecture. $X_{\operatorname{GKE}}$ is homeomorphic to $X_{can}$; in particular, $(X_{\operatorname{GKE}},d_{\operatorname{GKE}})$ is a compact metric space. This is proved for smooth minimal models of general type and when the Kodaira dimension is $1$, with further results for Kähler surfaces; the general case remains open.