Schoen's conjecture on products of curves dominating very general K3 surfaces

Let $S$ be a very general K3 surface. A dominant rational map from the product of two smooth projective curves to $S$ is a rational map $C\times D\dashrightarrow S$ whose image is dense in $S$. Schoen's conjecture. There can be no dominant rational map from the product of two smooth projective curves to $S$. This expectation concerns the nonexistence of isotrivial families of smooth curves dominating a very general K3 surface; the source attributes it, as a question, to Schoen.