Perturbation conjecture for Hamiltonian identity cycles

Let $\mathcal{F}^n=\{df=0\}$ be a degree $n$ Hamiltonian foliation, and let $\delta$ be a Hamiltonian identity cycle, meaning an identity cycle of this foliation. A degree $n$ perturbation is a perturbation within the space of degree $n$ polynomial foliations. Hamiltonian identity-cycle perturbation conjecture. There is a sufficiently small degree $n$ perturbation of $\mathcal{F}^n$ which either destroys $\delta$, or makes it a limit cycle. This is presented as a restricted version of the no-identity-cycle conjecture. The supplied text does not state whether it has been proved or disproved.