Scalar-flat Kähler metrics on deformations of minimal resolutions of quotient singularities

Let $\Gamma \subset {\rm{U}}(2)$ be a finite subgroup containing no complex reflections, and let $\widetilde{X}$ be the minimal resolution of $\mathbb{C}^2/\Gamma$. A small deformation of complex structure means a deformation of the complex structure on $\widetilde{X}$ sufficiently close to the original one. Scalar-flat Kähler ALE metric conjecture. All small deformations of the complex structure of $\widetilde{X}$ admit scalar-flat Kähler ALE metrics. This would extend the known constructions beyond the currently treated subspace of deformations. The result is known for some small deformations when $\Gamma$ is non-cyclic, while in the cyclic case it is not known except in cases overlapping with toric multi-Eguchi–Hanson and LeBrun negative-mass metrics.