The logarithmic diameter growth conjecture for Calabi–Yau degenerations

Let $\mathfrak{X}$ be a Calabi–Yau family as in Setup II that does not satisfy the equivalent conditions in Theorem. Logarithmic diameter growth conjecture. There exist constants $A,C>0$ such that the Ricci-flat Kähler metrics $\omega_t\in c_1(\mathfrak{L}|_{X_t})$ satisfy

$$\operatorname{diam}(X_t,\omega_t)\geqslant C^{-1}(-\log|t|)^A,$$

as $t\to0$. This is a quantitative form of the expected metric degeneration in the nontrivial large-complex-structure regime. The source presents it as an open problem attributed to Kontsevich–Soibelman.