Llarull's theorem on odd dimensional manifolds: the noncompact case

Let $(M,g^{TM})$ be an odd dimensional ($\dim M\geq 3$) connected oriented noncompact complete spin Riemannian manifold. Let $k^{TM}$ be the associated scalar curvature. Let $f:M\to S^{\dim M}(1)$ be a smooth area decreasing map which is locally constant near infinity and of nonzero degree. Suppose $k^{TM}\geq ({\dim M})({\dim M}-1)$ on the support of ${\rm d}f$, we show that $\inf(k^{TM})<0$. This answers a question of Gromov.