Killing-field classification conjecture for non-Kähler ALF gravitational instantons

Let $(\mathcal{M},g_{ab})$ be a non-Kähler ALF gravitational instanton admitting a non-vanishing Killing field with bounded norm. The conclusion of the Hermitian classification conjecture is that $(\mathcal{M},g_{ab})$ is one of Kerr, Chen--Teo, Taub-bolt, or Taub--NUT with the orientation opposite to the hyperkähler orientation.

Killing-field classification conjecture. The conclusion of the Hermitian classification conjecture holds for $(\mathcal{M},g_{ab})$.

The source states that the only open part is the case of toric instantons without special geometry; the conjecture is therefore open.