  Let $(M,F)$ be a connected (not necessarily compact) foliated manifold
carrying a complete Riemannian metric $g^{TM}$. We generalize Gromov's
$\mathrm{K}$-cowaist using the coverings of $M$, as well as defining a closely
related concept called $\widehat{\mathrm{A}}$-cowaist. Let $k^F$ be the
associated leafwise scalar curvature of $g^F = g^{TM}|_F$. We obtain some
estimates on $k^F$ using these two concepts. In particular, assuming that the
generalized $\mathrm{K}$-cowaist is infinity and either $TM$ or $F$ is spin, we
show that $\inf(k^F)\leq 0$.
