  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.
