Existence of a periodic-orbit cylinder in the generalized Chazy differential system

Let $q$ be a positive integer and set $k=q+1$. Consider the generalized Chazy differential system referred to as the generalized Chazy differential system.

Periodic-cylinder conjecture. For $k=q+1$ and $q$ a positive integer, the generalized Chazy differential system has an invariant cylinder foliated by periodic orbits.

The conjecture extends the computationally verified result for $1\leq q\leq 100$ and predicts the existence of such an invariant cylinder for every positive integer $q$. The stated context indicates that the result is supported by an algorithm based on the Sturm procedure, but does not establish it for all positive integers.