Foscolo–Haskins conjecture on cohomogeneity-one nearly Kähler six-manifolds

A compact nearly Kähler manifold is cohomogeneity one if it is equipped with an isometric action of a compact Lie group whose generic orbits have codimension one. Foscolo–Haskins conjecture. The only simply connected cohomogeneity-one compact nearly Kähler manifolds in dimension $6$ are the structures found by Foscolo and Haskins on $S^6$ and $S^3\times S^3$. These examples would classify the simply connected compact nearly Kähler six-manifolds admitting a cohomogeneity-one action; the conjecture concerns whether any others exist beyond the two known families.