Cautis–Logvinenko conjecture for the three-dimensional McKay correspondence

Let $G\subset \operatorname{SL}(3,\mathbb{C})$ be a finite subgroup, let $Y={G}\operatorname{-Hilb}(\mathbb{C}^3)$ be the $G$-Hilbert scheme, and let

$$\Psi\colon D^G(\mathbb{C}^3)\stackrel{\sim}{\longrightarrow}D(Y)$$

be the inverse of the Bridgeland–King–Reid derived equivalence. For a representation $\rho$ of $G$, write $\mathcal{O}_0\otimes\rho$ for the $G$-equivariant coherent sheaf supported at the origin. Cautis–Logvinenko conjecture. For each nontrivial irreducible representation $\rho$ of $G$, the object $\Psi(\mathcal{O}_0\otimes\rho)$ is a pure sheaf on $Y$. More precisely, there is a unique $k\in\{-1,0\}$ depending on $\rho$ such that

$$H^k\bigl(\Psi(\mathcal{O}_0\otimes\rho)\bigr)\neq 0.$$

This predicts a geometric realization of the three-dimensional McKay correspondence through the supports of the images of irreducible representations. The source does not state whether the conjecture has been resolved.