Deformation-equivalence conjecture for symplectic resolutions

Let $W$ be a symplectic variety, and let

$$\pi_i: Z_i \to W,\qquad i=1,2,$$

be two symplectic resolutions. Two proper morphisms are deformation equivalent if they occur as special fibers of deformations whose general fibers are isomorphisms.

Deformation-equivalence conjecture. The resolutions satisfy that $\pi_1$ is deformation equivalent to $\pi_2$, and $Z_1$ is diffeomorphic to $Z_2$.

The conjecture is proved when $W$ is projective, for nilpotent orbit closures, and under additional hypotheses; the supplied status evidence records a restricted proof, so the unrestricted conjecture is treated as open.