The Poisson-de Rham homology conjecture for affine symplectic resolutions

Let $\rho:\widetilde X\to X$ be a symplectic resolution, with $X$ affine, and let $\operatorname{\mathsf{HP}}_i^{DR}(X)$ denote the Poisson-de Rham homology of $X$. The affine symplectic-resolution conjecture. One has

$$\operatorname{\mathsf{HP}}_0(\mathcal O(X))\cong H^{\dim X}(\widetilde X),$$ $$\operatorname{\mathsf{HP}}_i^{DR}(X)\cong H^{\dim X-i}(\widetilde X)\quad\text{for all }i,$$

and

$$M(X)\cong\rho_*\Omega_{\widetilde X}.$$

The three assertions give a conjectural description of Poisson-de Rham homology for affine varieties admitting symplectic resolutions, and relate it to the ordinary cohomology of the resolution. The supplied text does not establish a resolution status for this bundled conjecture; the database therefore records it as open.