Universal-cover homology conjecture for positive intermediate curvature

Let $M^n$ be a closed manifold with $3\leq n\leq 7$, and let $\overline M$ be its universal cover. Assume that

$$H_n(\overline M,\mathbb{Z})=H_{n-1}(\overline M,\mathbb{Z})=\cdots=H_{n-m+1}(\overline M,\mathbb{Z})=0.$$

Universal-cover homology conjecture. Then $M$ does not admit a metric with positive $m$-intermediate curvature. The conjecture is proposed as a common generalization of the aspherical-manifold and generalized Geroch results; the source does not establish it in full.