Kollár's conjecture on Diophantine subsets of rational functions

Let $D\subseteq \mathbb{C}(z)$ be a Diophantine set. For each integer $n\geq 0$, write $\mathbb{C}[z]_n=\mathbb{C}^{n+1}$ for the polynomials of degree at most $n$. Kollár's conjecture. If, for infinitely many integers $n\geq 0$, the set $D$ contains a Zariski open subset of $\mathbb{C}[z]_n$, then $\mathbb{C}(z)-D$ is finite. This conjecture describes the expected rigidity of Diophantine subsets of $\mathbb{C}(z)$ that contain sufficiently large algebraic families of polynomials; the surrounding discussion presents it as an open conjecture related to Hilbert's tenth problem over $\mathbb{C}(z)$.