The compact nearly Kähler 3-symmetric-space conjecture

A compact nearly Kähler manifold is a nearly Kähler manifold whose underlying Riemannian manifold is compact. A 3-symmetric space is a homogeneous space associated with an automorphism of order 3.

Compact nearly Kähler conjecture. Every compact nearly Kähler manifold is a 3-symmetric space.

The paper identifies this as the remaining conjecture after resolving the homogeneous case in dimension 6. It states that the conjecture is still open, and notes that the homogeneous spaces presented in the article are consequently the only known compact, equivalently complete, examples in dimension 6.