Weak $K(\pi,1)$ conjecture for manifolds with boundary

Let $M^n$ be a compact $n$-manifold with non-empty boundary, and let its double be the closed manifold obtained by gluing two copies of $M^n$ along their common boundary. Let $\sigma(M^n)$ denote the Yamabe invariant of $M^n$. Weak $K(\pi,1)$ conjecture. If the double of $M^n$ is a $K(\pi,1)$ closed manifold, then

$$\sigma(M^n)\text{ is not positive}.$$

This is presented as a weaker boundary analogue of the $K(\pi,1)$ conjecture and would follow from that conjecture. The source does not provide a resolution, so the statement remains open.