Gray–Wolf conjecture on homogeneous strictly nearly Kähler spaces

A homogeneous space $M=G/H$ is equipped with a $G$-invariant Riemannian metric and its canonical almost-complex structure when $M$ is a 3-symmetric space: it admits a family of global isometries of order three, each having its associated point as an isolated fixed point. The metric is naturally reductive when, for a reductive decomposition $\mathfrak g=\mathfrak h\oplus\mathfrak m$, its inner product satisfies

$$([X,Y]_{\mathfrak m}|Z)=(X|[Y,Z]_{\mathfrak m})$$

for all $X,Y,Z\in\mathfrak m$. The canonical almost-complex structure is the endomorphism $J$ determined by

$$(\theta_m)_*=-\frac{1}{2}\operatorname{Id}_{T_mM}+\frac{\sqrt{3}}{2}J_m.$$

Gray–Wolf conjecture. Every homogeneous strictly nearly Kähler space is a naturally reductive 3-symmetric space equipped with its canonical almost-complex structure.