Ishii's moduli-space characterization conjecture for $G$-constellations

Let $G\subset\operatorname{GL}(2,\mathbb{C})$ be a finite subgroup, let $X=\mathbb{C}^2/G$ with boundary divisor $B$, and let $\mathcal{M}_{\theta}$ denote the moduli space of $\theta$-stable $G$-constellations for a generic stability parameter $\theta$. A resolution of singularities $\tilde{Y}\to X$ lies between the minimal and maximal resolution when it satisfies the ordering specified by those resolutions; the maximal resolution is the smooth variety with unique maximal coefficients satisfying the inequality in Definition. Ishii's conjecture. For any resolution of singularities $\tilde{Y}\to X$, $\tilde{Y}$ is isomorphic to $\mathcal{M}_{\theta}$ for some generic stability parameter $\theta$ if and only if $\tilde{Y}$ is between the minimal and maximal resolution of $\mathbb{C}^2/G$. This characterizes which resolutions arise as moduli spaces of stable $G$-constellations and extends the moduli-space form of the McKay correspondence from the special-linear setting to finite subgroups of $\operatorname{GL}(2,\mathbb{C})$.