Ginzburg's hyperkähler desingularization conjecture for reflection-group quotients

Let $G$ be a group generated by real reflections acting on a complex vector space $V$ and preserving a rational structure; more specifically, let $G$ be the Weyl group of some semisimple Lie group acting on its Cartan algebra. Consider the space $V\oplus V$ with the diagonal action of $G$, and let

$$X:=(V\oplus V)/G.$$

Ginzburg's conjecture. The quotient variety $X$ can be naturally desingularized, and this desingularization is holomorphically symplectic and admits a hyperkähler structure. The conjecture was disproved by counterexamples found in the paper; it is mentioned as a conjecture of V. Ginzburg that motivated the work.