Attractors and an analog of the Lichnérowicz conjecture for conformal foliations

We prove that each codimension q ≥ 3 conformal foliation (M,F) either is Riemannian or has a minimal set that is an attractor. If (M,F) is a proper conformal foliation that is not Riemannian then there exists a closed leaf that is an attractor. We do not assume that M is compact. Moreover, if M is compact then a non-Riemannian conformal foliation (M,F) is a (Conf(Sq), Sq)-foliation with a finite family of attractors, and each leaf of this foliation belongs to the basin of at least one attractor.