The natural-basis conjecture for homology of crepant resolutions of quotient singularities

Let $X=V/G$ be the quotient of a complex vector space $V$ by a finite subgroup $G\subset SL(V)$, and assume that $X$ admits a smooth crepant resolution $Y\to X$. Write $H_{\bullet}(Y,\mathbb{Q})$ for its total homology space. Natural-basis conjecture. The space $H_{\bullet}(Y,\mathbb{Q})$ admits a “natural” basis indexed by the conjugacy classes of elements $g\in G$. This is the homological form of the generalized McKay correspondence. It is established in dimensions two and three, while the general case remains open.