Metrics with positive Ricci eigenvalue sums on $\mathbb{T}^k\times \mathbb{S}^m$

We construct metrics with $k$-positive Ricci curvature on $\mathbb{T}^k\times \mathbb{S}^m$, for $k,m\geq2$, arbitrarily close to the standard flat-round product in the smooth topology. The two-torus case disproves Wolfson's conjecture that every closed manifold with two-positive Ricci curvature has