Butruille's conjecture on compact nearly Kähler manifolds

A nearly Kähler manifold is an almost Hermitian manifold whose canonical Hermitian connection has parallel and totally skew-symmetric torsion. A manifold is 3-symmetric if it is a homogeneous space admitting an automorphism of order three whose fixed-point set contains the isotropy subgroup.

Butruille's conjecture. Every compact nearly Kähler manifold is a 3-symmetric space.

The conjecture proposes that compact nearly Kähler geometry is completely described by 3-symmetric spaces. The source notes that the conjecture remains open; at the time of the paper, the known complete six-dimensional strict nearly Kähler examples were the homogeneous 3-symmetric spaces.