Rigidity conjecture for Calabi–Yau families with isolated singularities

Let $f\colon (\mathcal{X},\mathcal{L})\to S$ be a flat family of $n$-dimensional polarized smooth Calabi–Yau manifolds over a quasiprojective curve $S$. Let $\overline{S}$ be a projective smooth compactification of $S$ with a boundary point $0\in\overline{S}\setminus S$. Suppose that the family extends to a flat family $\overline{\mathcal{X}}\to\overline{S}$ such that the total space $\overline{\mathcal{X}}$ is smooth and the singular fiber $X_0=f^{-1}(0)$ has only isolated singularities. Rigidity conjecture. Under these assumptions, the family $f\colon (\mathcal{X},\mathcal{L})\to S$ is rigid. Known nonrigid Calabi–Yau families tend to have singular fibers whose singular loci have positive dimension; the conjecture proposes that isolated singularities in a suitable compactification obstruct non-rigidity. The supplied text does not state whether the conjecture has been resolved.