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 $\mathbb{Ca}^3$. Together with our earlier work, this concludes the classification of simply-connected homogeneous spaces that admit a homogeneous metric with strongly positive curvature.