Finiteness conjecture for symplectic resolutions

Let $W$ be a symplectic variety. A symplectic resolution is a resolution of $W$ on which the symplectic form extends to a symplectic form. Two resolutions are non-isomorphic when they are not isomorphic as resolutions of $W$.

Finiteness conjecture. The variety $W$ has at most finitely many non-isomorphic symplectic resolutions.

The conjecture is verified for nilpotent orbit closures and for some four-dimensional and quotient examples; the supplied text leaves the general statement open.