Weil–Petersson completeness conjecture for Calabi–Yau threefold moduli

A polarized Calabi–Yau manifold is a pair $(X,\omega)$ consisting of a compact algebraic manifold $X$ with zero first Chern class and a Kähler form $\omega\in H^2(X,\mathbb Z)$. Let ${\mathcal M}$ be the universal deformation space of $(X,\omega)$, equipped with its Weil–Petersson metric $G_{WP}$. In complex dimension three, ${\mathcal M}$ is a projective special Kähler manifold. Weil–Petersson completeness conjecture. If the moduli space ${\mathcal M}$ of a Calabi–Yau threefold is complete with respect to the Weil–Petersson metric, then it is locally symmetric. The question concerns whether completeness of the Weil–Petersson metric can occur; in the threefold case, the conjecture predicts that any complete example must have locally symmetric geometry.