Nonexistence conjecture for analytic first integrals of even Liénard systems

Let $E(x)$ be an even polynomial of degree at least $4$, and consider the system

$$\dot{x}=y-E(x),\qquad \dot{y}=-x.$$

Nonexistence conjecture. There is no global analytic first integral for this system on $\mathbb{R}^2$. The claim concerns global analytic integrability of planar Liénard systems; the source introduces it as a conjecture, and no resolution is supplied there.