$\mathrm{K}$-cowaist on complete foliated manifolds

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 the $\widehat{\mathrm{A}}$-cowaist. Let $k^F$ be the associated leafwise scalar curvature of $g^F=g^{TM}{\mathrm{\mid }}_F$. 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 $\inf (k^F)\le 0$.