Positive intermediate Ricci curvature on cohomogeneity one manifolds in low dimensions

We explore existence of invariant metrics with positive intermediate Ricci curvature on closed, low-dimensional cohomogeneity one manifolds. For a certain cohomogeneity one $\mathsf{S}\mathsf{p}\mathsf{i}\mathsf{n}(4)$-action on $S^3\times \mathbb{C}{\mathrm{P}}^2$, we construct an invariant metric with positive 4th-intermediate Ricci curvature and show it cannot admit one with positive 3rd-intermediate Ricci curvature. We further establish similar symmetry obstructions to positive curvature for $S^3\times S^3$, $S^3\times S^4$, and several families of cohomogeneity one manifolds.