This is one of these questions where it's tempting to just leave it at the title, but let me try to define the objects in question.

A conical symplectic resolution is a projective resolution of singularities $X \to Y$ such that

• $X$ is algebraically symplectic,
• $Y$ is affine, and
• there are compatible $\mathbb{G}_m$-actions on the two varieties which make $Y$ into a cone and act on the symplectic form with positive weight $n$.

Examples include the Springer resolution, a minimal resolution of a rational double point, the Hilbert scheme of points in that space (via the Hilbert-Chow resolution), a hypertoric variety or a Nakajima quiver variety.

All of these spaces have something in common: they are Mori dream spaces.

Thus, I am inclined to wonder:

Are all conical symplectic resolutions Mori dream spaces? Or am I just not original enough to come up with counter examples?