The complete-intersection quotient conjecture for projective crepant resolutions

Let $G$ be a finite subgroup of $\operatorname{SL}(d,\mathbb{C})$ such that $\mathbb{C}^d/G$ is minimally embeddable as a complete intersection in an affine space $\mathbb{C}^r$, where $r\geq d+1$. A complete-intersection quotient conjecture asserts that, for all $d\geq 2$, the quotient space $\mathbb{C}^d/G$ admits a crepant projective desingularization. This proposes a significant class of Gorenstein quotient singularities for which the higher-dimensional McKay correspondence can be realized using full projective crepant resolutions; the source presents it as a belief motivated by known three-dimensional existence results and the scarcity of higher-dimensional examples.