Positive intermediate curvatures and Ricci flow

We show that, for any $n\ge 2$, there exists a homogeneous space of dimension $d=8n-4$ with metrics of ${\mathrm{R}\mathrm{i}\mathrm{c}}_{\frac{d}{2}-5}>0$ if $n\mathrm{\ne }3$ and ${\mathrm{R}\mathrm{i}\mathrm{c}}_6>0$ if $n=3$ which evolve under the Ricci flow to metrics whose Ricci tensor is not $(d-4)$-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking in the case of $n=2$.