Kollár's conjecture on diophantine subsets of the rational function field

Let $D$ be a diophantine subset of $\mathbb{C}\left(z\right)$, and for each $n\in\mathbb{Z}_{\geq 1}$ let $\mathbb{C}\left[z\right]_n$ be the $(n+1)$-dimensional $\mathbb{C}$-vector subspace of $\mathbb{C}\left[z\right]$ spanned by $\{1,x,\cdots,x^n\}$. Endow $\mathbb{C}\left[z\right]_n$ with the Zariski topology inherited from $\mathbb{C}^{n+1}$. Kollár's conjecture. If $D$ contains a Zariski open subset of $\mathbb{C}\left[z\right]_n$ for infinitely many $n\in\mathbb{Z}_{\geq 1}$, then

$$\mathbb{C}\left(z\right)\setminus D$$

is finite. The conjecture concerns the tension between diophantine definability and largeness in the spaces of polynomials; the supplied context notes that it is incompatible with the diophantineness of Campana points in $\mathbb{C}\left(z\right)$ except in trivial cases.