Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen

We prove that there exist ${\mathsf{S}\mathsf{U}}_3$-invariant metrics on Aloff-Wallach spaces $W_{k_1,k_2}^7$, as well as ${\mathsf{S}\mathsf{U}}_5$-invariant metrics on the Berger space $B^{13}$, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.