Disconnectedness conjecture for finite abelian G-Hilbert schemes

Let $G$ be a finite abelian subgroup of ${\operatorname{GL}}(n,\Bbbk)$, and let ${G}\operatorname{-Hilb}$ denote the associated $G$-Hilbert scheme. Disconnectedness conjecture. There exists a finite abelian subgroup $G\subset {\operatorname{GL}}(n,\Bbbk)$ such that

$${G}\operatorname{-Hilb}$$

is disconnected. This conjecture arises from the fact that toric Hilbert schemes can be disconnected; the source does not provide a resolution.