Hermitianity conjecture for ALF gravitational instantons

An ALF gravitational instanton is a gravitational instanton with asymptotically locally flat geometry, including the ALF families discussed in the source. A Riemannian four-manifold is Hermitian when its metric is compatible with an integrable complex structure.

ALF Hermitianity conjecture. Every ALF gravitational instanton is Hermitian.

This is stronger than the preceding classification conjectures and is stated as an analogue of earlier conjectures on gravitational instantons. The source does not provide a resolution, so the conjecture remains open.