Non-existence conjecture for Hermitian non-Kähler ALE gravitational instantons

A Hermitian non-Kähler ALE gravitational instanton is a Hermitian non-Kähler ALE Ricci-flat 4-manifold, with no restriction on its finite structure group. The Eguchi–Hanson metric with reversed orientation is the distinguished exceptional example.

Non-existence conjecture. There exist no Hermitian non-Kähler ALE gravitational instantons, without any restrictions on the structure group, except for the Eguchi–Hanson metric with reversed orientation.

The conjecture would extend the paper’s classification results beyond structure groups contained in $SU(2)$. The obstruction to proving it is the more complicated classification of log del Pezzo surfaces with one $U(2)$ singularity and a holomorphic vector field with matching weights; the source does not state a resolution.