  Let $Y^n$ be a closed oriented Schoen-Yau-Schick, enlargeable or
$4$-dimensional aspherical manifold. In this note we prove generalized band
width estimates for $Y^n\times[-1,1]$, and collar width estimates for
$Y^n\times S^1$, with a point removed, in the presence of positive lower scalar
curvature bounds and for $n\leq6$. These results are inspired by recent work of
Cecchini and Zeidler in the spin setting and include the mean curvature of the
boundary into the equation. We use results and techniques of Chodosh and Li,
Gromov and Zhu regarding so called $\mu$-bubbles, which in short are
hypersurfaces minimizing a warped volume functional.
