The coherent component conjecture for McKay quiver moduli

Let $G\subset \operatorname{GL}(n,\Bbbk)$ be a finite subgroup. For all $\theta$ in the rational character space, let $\mathcal{M}_\theta=Z{\operatorname{/\tslash}}_\theta H$ be the GIT quotient of an affine scheme $Z$ by an algebraic group $H$, and let $z\in Z$ be the quiver representation defined by the $G$-orbit of $(1,\dots,1)\in \mathbb{A}^n_\Bbbk$. Let $V$ be the subscheme of $Z$ obtained as the closure of the $(\Bbbk^*)^n\times H$-orbit of $z$. For generic $\theta\in H^*\otimes \mathbb{Q}$, the GIT quotient

$$Y_\theta:=V{\operatorname{/\tslash}}_\theta H$$

is a reduced irreducible component of $\mathcal{M}_\theta$ and admits a projective birational morphism

$$Y_\theta\longrightarrow Y_{\bf 0}\cong \mathbb{A}^n_\Bbbk/G$$

obtained by variation of GIT quotient. Coherent component conjecture. The component obtained by taking the closure of the torus-and-group orbit of the distinguished representation should have these reduced, irreducible, and birational properties for generic stability parameter. This conjecture extends the construction beyond the abelian case and predicts a canonical component of the moduli space governing the birational geometry of the quotient singularity.