Classification of complete cohomogeneity-one nearly Kähler structures on four six-manifolds

The relevant compact six-manifolds are $S^6$, $S^3\times S^3$, $CP^3$, and $S^2\times S^4$.

Classification conjecture. The Main Theorem yields all inhomogeneous complete cohomogeneity-one nearly Kähler structures. In particular, $S^2\times S^4$ admits no cohomogeneity-one nearly Kähler structure, and $CP^3$ admits only its homogeneous one.

The Main Theorem proves existence of inhomogeneous structures on $S^6$ and $S^3\times S^3$; the asserted exhaustive classification and the claims for $CP^3$ and $S^2\times S^4$ remain open.