  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 complete
metric with nonnegative scalar curvature. In particular, a special case of our
result answers a question of Gromov.
  More generally, we generalize the partial classification result of Chodosh,
Li, and Liokumovich to the non-compact domination case with our newly-developed
technique.
  Our result unifies all previous results of this type, and confirms the
validity of Gromov's non-compact domination conjecture for closed aspherical
manifolds of dimensions 3, 4, and 5.
