Tosatti's homeomorphism conjecture for Gromov-Hausdorff limits of Calabi-Yau fibrations

Let $M$ be a projective manifold with a semiample line bundle $L$ inducing a holomorphic map $f:M\to X$, and let $X^\circ\subset X$ be a Zariski open subset over which $f$ is a holomorphic submersion. For Ricci-flat Kähler metrics $\omega_t$ on $M$ with $[\omega_t]=c_1(L)+tc_1(A)$, where $A$ is ample and $t\to 0$, let $(Z,d_Z)$ be the Gromov-Hausdorff limit, equivalently the metric completion of $(X^\circ,\omega_{can})$. Tosatti's conjecture. The Gromov-Hausdorff limit $(Z,d_Z)$ is homeomorphic to $X$, and the set $X\setminus X^\circ$ has Hausdorff codimension at least two in $(Z,d_Z)$. This conjecture concerns the global topology and metric singularities of collapsing Ricci-flat metrics; the local smooth convergence to the canonical metric is known, and the stated global identification remains an important problem in the general collapsing setting.