Uniform boundedness conjecture for zeros of pseudoabelian integrals

Consider a one-parameter family of planar real foliations defined by

$$\theta+\varepsilon\omega=0,$$

where $\theta$ is a closed rational $1$-form and $\omega$ is a rational $1$-form admissible for $\theta$, meaning that $\operatorname{Poles}(\omega)\subseteq\operatorname{Poles}(\theta)$. For ovals $L_t\subseteq\{f=t\}$, write the associated Poincaré–Pontryagin integral as

$$I(t)=\oint_{L_t}\omega.$$

Uniform boundedness conjecture. For any pair of rational $1$-forms $(\theta,\omega)$ on the real plane $\mathbb R^2$ of degrees $n,m$ respectively, such that $d\theta=0$ and $\operatorname{Poles}(\omega)\subseteq\operatorname{Poles}(\theta)$, the number of isolated real zeros of $I(t)$ is bounded by a constant $N=N(n,m)$ depending only on $n$ and $m$.

Such a bound would translate directly, via the Poincaré–Pontryagin criterion, into uniform bounds for limit cycles arising from polynomial perturbations of Darbouxian integrable systems. The source presents this as a conjecture believed by most experts; no resolution is supplied.