Nonexistence of centers for polynomial differential equations without linear terms

Let $N\geq 3$, let $A_j(t)$, for $j=2,3,\ldots,N-1$, be polynomial functions, and consider the differential equation

$$\dot{z}=z^{N}+A_{N-1}(t)z^{N-1}+\cdots+A_{3}(t)z^{3}+A_{2}(t)z^{2}.$$

A solution is a center at the origin if nearby solutions have the corresponding periodicity property around $z=0$. Nonexistence conjecture. The solution $z=0$ is not a center for this differential equation. The conjecture asserts that polynomial differential equations without linear terms do not have centers at the origin when their coefficients are polynomial functions of $t$.