Homogeneity of cohomogeneity-one nearly Kähler metrics on P^3

Determine whether every cohomogeneity-one nearly Kähler metric on P^3 is homogeneous.

Background

The paper discusses the special role of dimension six, where the nearly Kähler condition implies the Einstein condition, and describes a non-integrable almost complex structure on P3 arising from the twistor fibration.

In that setting, Foscolo and Haskins are cited as conjecturing that all cohomogeneity-one nearly Kähler metrics on P3 are homogeneous. The statement is presented as a conjecture rather than as a theorem, so the classification question remains open.

References

It is interesting to point out that Foscolo and Haskins conjectured that every cohomogeneity-one nearly Kähler metric on P3 is homogeneous.

— Inhomogeneous Einstein metrics on complex projective spaces  (2608.16880 - Cao-Labora et al., 17 Aug 2026) in Remark 2.14, Section 2.3.3 (remark labeled “nonNK”)