Anosov's conjecture on identical cycles of generic polynomial foliations

Let $\mathcal{A}_n$ be the space of foliations of $\mathbb{C}^2$ defined by polynomial vector fields of degree at most $n$ with coprime components, equipped with its natural topology. An identical cycle on a leaf $L$ is a non-trivial element of the free homotopy group of $L$ whose holonomy is the identity.

Anosov's conjecture. A generic polynomial foliation has no identical cycles.

If true, this would imply that almost all leaves of a generic polynomial foliation are topological discs, paralleling the known result for generic analytic foliations. The conjecture is presented as unresolved in the source.