Is there a polynomial function $P:\mathbb{R}^2 \to \mathbb{R}$ with the following property?

For sufficiently large $c>0$, $P^{-1}(c)$ is a simple closed curve $\gamma_c$, homeomorphic to $S^1$, but the centroid $e_c$ of the interior of $\gamma_c$ does not converge to any point of $\mathbb{R}^2$ as $c$ goes to $+\infty$.

Motivation: The answer is negative if we consider this question for polynomials $p:\mathbb{R} \to \mathbb{R}$ whose eventual level sets are $2$-pointed set, i.e. $S^0$.(Namely a polynomial of even degree). The motivation comes from line -3, item III, page 4 of Taghavi - On periodic solutions of Liénard equations, which can be generalized to every even degree polynomial with one variable.