Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

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