The essential-skeleton metric limit conjecture

Let $\mathfrak{X}\subset\mathbb{CP}^N\times\Delta$ be a family of Calabi–Yau $n$-folds with

$$0<\dim_{\mathbb{R}}Sk(\mathfrak{X})<n.$$

Equip $X_t$ for $t\neq0$ with the Ricci-flat Kähler metrics $\omega_t\in c_1(\mathfrak{L}|_{X_t})$ and define

$$\widetilde{\omega}_t=\operatorname{diam}(X_t,\omega_t)^{-2}\omega_t.$$

Essential-skeleton metric limit conjecture. As $t\to0$, $(X_t,\widetilde{\omega}_t)$ converges in the Gromov–Hausdorff topology to a compact metric space $(Z,d)$ homeomorphic to the essential skeleton $Sk(\mathfrak{X})$. The conjecture extends known gluing constructions for certain $K3$ families, while the general collapsing behavior remains open.