  An almost quaternion-Hermitian structure on a Riemannian manifold
$(M^{4n},g)$ is a reduction of the structure group of $M$ to
$\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \mathrm{SO}(4n)$. 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
$\mathrm{SO}(7)/\mathrm{U}(3)$.
