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?

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 (relative) Mori dream spaces. (For a definition of "relative Mori dream space," see this paper).

Thus, I am inclined to wonder:

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