Homeomorphism and codimension conjecture for Calabi–Yau collapse limits

Let $M^m$ be a projective Calabi–Yau manifold with $K_M\cong\mathcal{O}_M$, equipped with a fiber space structure $f:M\to N$, where $N^n$ is a normal projective variety and $0<n<m$. Let $D\subset N$ be the union of the singularities of $N$ and the critical values of $f$ on $N^{\rm reg}$, set $N^\circ=N\backslash D$ and $M^\circ=M\backslash f^{-1}(D)$, and let $(Z,d_Z)$ be the metric completion of $(N^\circ,\omega_{\rm can})$, where $\omega_{\rm can}$ is the canonical Kähler metric determined by the collapsing Ricci-flat metrics. Calabi–Yau collapse conjecture. In the Calabi–Yau setup, the Gromov–Hausdorff limit is homeomorphic to $N$. Furthermore, $Z\backslash N^\circ$ has real Hausdorff codimension at least $2$ inside $(Z,d_Z)$. This concerns the topology and metric size of the singular set in the Gromov–Hausdorff limit of collapsing Ricci-flat Kähler metrics; the convergence to $(Z,d_Z)$ is known, but the asserted homeomorphism and codimension statement remain open in general.