Attractors and an analog of the Lichnérowicz conjecture for conformal foliations

We prove that each codimension $q\ge3$ conformal foliation $(M,\mathcal F)$ either is Riemannian or has a minimal set that is an attractor. If $(M,\mathcal F)$ 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 $(M,\mathcal F)$ is a $(\operatorname{Conf}(S^q),S^q)$-foliation with a finite family of attractors, and each leaf of this foliation belongs to the basin of at least one attractor.