Generalised Geroch conjecture with boundary

Let $n\geq 2$ be such that the generalised Geroch conjecture without boundary holds for all $n$-manifolds. Let $M$ be an $n$-manifold with non-empty boundary, and form $M\# T^n$ by attaching a torus in the interior. A boundary is mean convex when its mean curvature is non-negative at every boundary point. Generalised Geroch conjecture with boundary. The manifold $M\# T^n$ does not admit a complete Riemannian metric of positive scalar curvature with mean convex boundary. This boundary version is conditional on the corresponding conjecture without boundary and is presented as a consequence in the source; its status as a standalone general statement is therefore open.