  In this paper, we prove that the foliated Rosenberg index of a possibly
noncompactly enlargeable, spin foliation is nonzero. It generalizes our
previous result. The difficulty brought by the noncompactness is reflected in
the infinite dimensionality of some vector bundles which, fortunately, can be
reduced to finite dimensional vector bundles by the idea of relative index
theorem and $KK$-equivalence between the $C^\ast$-algebra of compact operators
and $\mathbb{C}$.
