  Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension
$n$, and $F\subseteq TM$ be an integrable subbundle of $TM$. Let
$g^F=g^{TM}|_{F}$ be the restricted metric on $F$ and $k^F$ be the associated
leafwise scalar curvature. Let $f:M\to S^n(1)$ be a smooth area decreasing map
along $F$, which is locally constant near infinity and of non-zero degree. We
show that if $k^F> {\rm rk}(F)({\rm rk}(F)-1)$ on the support of ${\rm d}f$,
and either $TM$ or $F$ is spin, then $\inf (k^F)<0$. As a consequence, we prove
Gromov's sharp foliated $\otimes_\varepsilon$-twisting conjecture. Using the
same method, we also extend two famous non-existence results due to Gromov and
Lawson about $\Lambda^2$-enlargeable metrics (and/or manifolds) to the foliated
case.
