Landau–Ginzburg compactification conjecture for Fano varieties

Let $X$ be a smooth Fano variety with very ample anticanonical class. A Landau–Ginzburg model for $X$ is a Laurent polynomial whose constant coefficient series coincides with the regularised quantum period of $X$. A smooth log Calabi–Yau compactification is a compactification of this model to a smooth projective morphism $\mathcal{W}\to\mathbb{P}^1$ satisfying $-K_{\mathcal{W}}\sim\mathsf{f}^{-1}(\infty)$.

Landau–Ginzburg compactification conjecture. $X$ admits a Landau–Ginzburg model with a smooth log Calabi–Yau compactification.

The conjecture is known for smooth del Pezzo surfaces, smooth Fano threefolds, and smooth Fano complete intersections in projective space. The supplied text gives no broader resolution status.