  We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci
curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved
homogeneous spaces. Using these examples, we demonstrate that the Hopf
conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and
Hsiang-Kleiner theorem for positively curved manifolds do not hold in the
$\mathrm{Ric}_2>0$ setting. These observations indicate that the class of
manifolds with $\mathrm{Ric}_2>0$ is vastly different from the class of
positively curved manifolds.
