The groupoid isomorphism conjecture for Kleinian singularities

Let $V$ be a two-dimensional complex vector space and let $\Gamma$ be a finite subgroup of $SL(2,\mathbb{C})$, so that $V/\Gamma$ is a Kleinian singularity. Let $\operatorname{PIso}(\mathbb{C}[V/\Gamma])$ denote the groupoid of Poisson isomorphisms and let $\operatorname{Iso}(\mathbb{C}[V/\Gamma])$ denote the groupoid of algebra isomorphisms. Groupoid isomorphism conjecture. There is an isomorphism of groupoids

$$\operatorname{PIso}(\mathbb{C}[V/\Gamma])\cong\operatorname{Iso}(\mathbb{C}[V/\Gamma]).$$

The conjecture concerns whether Poisson and ordinary isomorphisms of the function algebra on a Kleinian singularity encode the same groupoid structure. The source formulates it without giving evidence of resolution.