Homogeneous almost quaternion-Hermitian manifolds

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{S}\mathrm{p}(n)\mathrm{S}\mathrm{p}(1)\subset \mathrm{S}\mathrm{O}(4n)$. In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characteristic is either a Wolf space, or ${\mathbb{S}}^2\times {\mathbb{S}}^2$, or the complex quadric $\mathrm{S}\mathrm{O}(7)\mathrm{/}\mathrm{U}(3)$.