The Hamiltonian-reduction implication for Condition (GIT)

Let $V$ be a linear representation of a complex reductive algebraic group $G$ and let

$$\mu:T^*V\to\mathfrak{g}:=\operatorname{Lie}G$$

be the associated moment map. Set $V:=\mu^{-1}(0)$. For a generic $\theta\in G^\vee_{\mathbb{Q}}$, suppose that the VGIT morphism $X_\theta\to X_0$ is a crepant resolution and that the associated linearisation map $L_C$ is an isomorphism. Hamiltonian-reduction conjecture. Under these hypotheses, Condition~ holds. The claim concerns Hamiltonian reductions associated to representations of reductive groups, including Nakajima quiver varieties and hypertoric varieties. It was anticipated without proof in the cited work, and the source gives no resolution status.