The tautological-sheaf basis conjecture for quotient resolutions

Let $G\subset\operatorname{SL}(n,\mathbb C)$ be finite, let $X=\mathbb C^n/G$, and let $f\colon Y\to X$ be a resolution. For each irreducible representation $\rho\colon G\to\operatorname{GL}(V_\rho)$, define

$$\mathcal F'_\rho:=\operatorname{Hom}(V_\rho,\mathcal O_{\mathbb C^n})^G,$$

and let $\mathcal F_\rho$ be its torsion-free birational transform on $Y$.

Tautological-sheaf basis conjecture. Under appropriate circumstances, the tautological sheaves $\mathcal F_\rho$ form a $\mathbb Z$-basis of $K_0(\operatorname{Coh}Y)$; their Chern classes should yield a $\mathbb Z$-basis of $H^*(Y,\mathbb Z)$. More strongly, the $\mathcal F_\rho$ should form a $\mathbb Z$-basis of the derived category $D^b(\operatorname{Coh}Y)$.

The conjecture is intended as a sheaf-theoretic and derived-categorical refinement of the McKay correspondence. The source describes it as a speculative framework and does not specify the circumstances or prove the general assertion.