Cautis–Logvinenko purity conjecture for objects on the G-Hilbert scheme

Let $G$ be a finite, non-abelian subgroup of $\operatorname{SL}(3,\mathbb{C})$, let $C$ be the chamber defining ${G}\operatorname{-Hilb}(\mathbb{C}^3)$, and let $S_\rho$, $L_\rho$, and $\Psi(S_\rho)$ be the objects and line bundle associated with a nontrivial representation $\rho$ of $G$. Write $Z_\rho$ for the unstable locus of the wall $\overline{C}\cap S_\rho^\perp$. Cautis–Logvinenko's purity conjecture, with the wall description. The object $\Psi(S_\rho)$ is a pure sheaf in degree $0$ if and only if $\overline{C}\cap S_\rho^\perp$ is a wall of the chamber $C$, in which case

$$\Psi(S_\rho)\cong L_\rho^{\vee}\vert_{Z_\rho}.$$

This refines the cited conjecture that the objects $\Psi(S_\rho)$ arising from nontrivial representations are pure sheaves. The proposed wall criterion and identification with the restriction to the unstable locus are motivated by Proposition 1.3 in the paper, but the parser supplies no evidence that the conjecture has been resolved.