On the Craw-Ishii conjecture

In [2], Craw and Ishii proved that for a finite abelian group G⊂SL3(C) every (projective) relative minimal model of C3/G is isomorphic to the fine moduli space Mθ of θ-stable G-constellations for some GIT parameter θ. In this article, we conjecture that the same is true for a finite group G⊂GL3(C) if a relative minimal model Y of X=C3/G is smooth. We prove this for some abelian groups.