The weight-generating-function conjecture for Poisson-de Rham homology

Let $X$ be a conical Poisson variety with a symplectic resolution, and let $d$ be the degree of its generic symplectic form. Let $\widetilde X$ be the resolution and, for each $i$, let $\phi(i,j)\in\mathbf Z_{\geq0}$ be the exponents describing the Jordan blocks of the cokernel of the map from fiberwise closed $i$-forms to $H^i(\widetilde X)\otimes\mathbf C[t]$. Write $h(\operatorname{\mathsf{HP}}_*^{DR}(X);x,y)$ for the corresponding bigraded generating function. The weight-generating-function conjecture. One has

$$h(\operatorname{\mathsf{HP}}_*^{DR}(X);x,y)=y^{-d\cdot\dim X/2}\sum_{i,j}x^i y^{d\cdot\phi(\dim X-i,j)}.$$

The conjecture relates the bigraded Poisson-de Rham homology to the Gauss–Manin-theoretic defect measured by the integers $\phi(i,j)$; in particular, it predicts that the homology is supported only in weights divisible by $d$. The supplied text gives no resolution evidence, so the conjecture remains open.