  Using $\mu$-bubbles, we prove that for $3 \le n \le 7$, the connected sum of
a Schoen-Yau-Schick $n$-manifold with an arbitrary manifold does not admit a
complete metric of positive scalar curvature.
  When either $3 \le n \le 5$, $1 \le m \le n-1$ or $6 \le n \le 7$, $m \in
\{1, n-2, n-1\}$, we also show the connected sum $(M^{n-m}\times \mathbb{T}^m)
\# X^n$ where $X$ is an arbitrary manifold does not admit a metric of positive
$m$-intermediate curvature. Here $m$-intermediate curvature is a new notion of
curvature introduced by Brendle, Hirsch and Johne interpolating between Ricci
and scalar curvature.
