  Let $M$ be an orientable connected $n$-dimensional manifold with
$n\in\{6,7\}$ and let $Y\subset M$ be a two-sided closed connected
incompressible hypersurface which does not admit a metric of positive scalar
curvature (abbreviated by psc). Moreover, suppose that the universal covers of
$M$ and $Y$ are either both spin or both non-spin. Using Gromov's
$\mu$-bubbles, we show that $M$ does not admit a complete metric of psc. We
provide an example showing that the spin/non-spin hypothesis cannot be dropped
from the statement of this result. This answers, up to dimension $7$, a
question by Gromov for a large class of cases. Furthermore, we prove a related
result for submanifolds of codimension two. We deduce as special cases that, if
$Y$ does not admit a metric of psc and $\dim(Y) \neq 4$, then $M :=
Y\times\mathbb{R}$ does not carry a complete metric of psc and $N := Y \times
\mathbb{R}^2$ does not carry a complete metric of uniformly psc provided that
$\dim(M) \leq 7$ and $\dim(N) \leq 7$, respectively. This solves, up to
dimension $7$, a conjecture due to Rosenberg and Stolz in the case of
orientable manifolds.
