The generalized Geroch conjecture for connected sums with tori

Let $M$ be a manifold of dimension $m$, and let $T^m$ denote the $m$-dimensional torus. Generalized Geroch conjecture. There is no complete metric of positive scalar curvature on

$$T^m \# M.$$

This generalizes the Geroch conjecture, which concerns whether the torus $T^m$ admits a metric of positive scalar curvature. The claim is known for $M$ a point by work of Schoen–Yau in dimensions $3\leq m\leq 11$ and independently of Gromov–Lawson in all dimensions, but the stated connected-sum version is presented here as a conjecture.