The primitive Hodge number conjecture for toric Landau–Ginzburg models

Let $X$ be a smooth Fano variety of dimension $n$, and let $Y$ be a Calabi–Yau compactification of its toric Landau–Ginzburg model. Let $k_Y$ be the number of irreducible components of all reducible fibres of $Y$ minus the number of reducible fibres, and let $h_{pr}^{p,q}(X)$ denote the primitive Hodge numbers of $X$. A primitive Hodge number conjecture asserts that

$$h_{pr}^{1,n-1}(X)=k_Y.$$

This relates a numerical invariant of the compactified Landau–Ginzburg model to the primitive Hodge structure of the Fano variety. The source gives no resolution status.