Poisson-cohomology conjecture for symplectic resolutions

Let $W$ be a symplectic variety, let $Z \to W$ be a symplectic resolution, and let $HP^*(W)$ denote the Poisson cohomology of $W$.

Poisson-cohomology conjecture. There is an isomorphism of vector spaces

$$H^*(Z,\mathbb C) \simeq HP^*(W).$$

In particular, $HP^*(W)$ is finite-dimensional.

The claim is known when $W$ is smooth, and the supplied text gives partial verification for quotient varieties in low cohomological degrees; the general statement remains open.