  We prove that there exist $\mathsf{SU}_{3}$-invariant metrics on
Aloff-Wallach spaces $W^7_{k_1, k_2}$, as well as $\mathsf{SU}_{5}$-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.
