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We prove for $n\in\{3,4,5\}$ that the connected sum of a closed aspherical$n$-manifold with an arbitrary non-compact manifold does not admit a completemetric with nonnegative scalar curvature. In particular, a special case of ourresult answers a question of Gromov. In addition, we prove a correspondingrigidity statement for the classification result of Chodosh, Li, andLiokumovich on closed manifolds admitting positive scalar curvature.

Positive scalar curvature metric and aspherical summands