Positive ${\mathrm{R}\mathrm{i}\mathrm{c}}_2$ curvature on products of spheres and their quotients via intermediate fatness

We construct metrics of positive $2^{\mathrm{n}\mathrm{d}}$ intermediate Ricci curvature, ${\mathrm{R}\mathrm{i}\mathrm{c}}_2>0$, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including ${\mathbb{S}}^6\times {\mathbb{S}}^7$, ${\mathbb{S}}^7\times {\mathbb{S}}^7$ and all their simply connected isometric quotients. In particular, we obtain infinitely many examples in dimension 13. We also produce infinitely many non-simply connected spaces with ${\mathrm{R}\mathrm{i}\mathrm{c}}_2>0$ in dimensions 13 and 14, including ${\mathbb{R}\mathbb{P}}^6\times {\mathbb{R}\mathbb{P}}^7$ and ${\mathbb{R}\mathbb{P}}^7\times {\mathbb{R}\mathbb{P}}^7$, which cannot admit a metric of positive sectional curvature. The main new idea is a generalization of the concept of fatness which ensures the existence of ${\mathrm{R}\mathrm{i}\mathrm{c}}_2>0$ metrics on the total space of certain homogeneous bundles.