Landau–Ginzburg Hodge-number equality for Fano-type models

Let $(Y,w)$ be a Landau–Ginzburg model of Fano type, with Landau–Ginzburg Hodge numbers $h^{p,q}(Y,w)$, $f^{p,q}(Y,w)$, and $i^{p,q}(Y,w)$ as defined in the paper. Landau–Ginzburg Hodge-number conjecture. For every $p,q$ there are equalities

$$h^{p,q}(Y,w)=f^{p,q}(Y,w)=i^{p,q}(Y,w).$$

This conjecture refines the equality of the sums of the three kinds of Hodge numbers. The source notes that the assertion fails for the Landau–Ginzburg model of $\mathbb P^2$, so the conjecture as stated is refuted.