Gray and Wolf's conjecture on homogeneous nearly Kähler manifolds

A nearly Kähler manifold is a Riemannian manifold with an almost Hermitian structure whose canonical Hermitian connection satisfies the nearly Kähler condition. A 3-symmetric space is a homogeneous space associated with an automorphism of order 3, equipped with its canonical almost complex structure.

Gray and Wolf's conjecture. Every nearly Kähler homogeneous manifold is a 3-symmetric space equipped with its canonical almost complex structure.

The paper gives a positive answer to this conjecture by classifying six-dimensional homogeneous nearly Kähler manifolds; the four examples are $S^3 \times S^3$, $\mathbb{C}P^3$, the flag manifold $\mathbb{F}^3$, and $S^6$. The stated conjecture is therefore solved in the paper.