Flag manifolds with strongly positive curvature

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds $W^6$, $W^{12}$ and $W^{24}$, which are respectively the manifolds of complete flags in ${\mathbb{C}}^3$, ${\mathbb{H}}^3$ and ${\mathbb{C}\mathbb{a}}^3$. Together with our earlier work, this concludes the classification of simply-connected homogeneous spaces that admit a homogeneous metric with strongly positive curvature.