Density conjecture for non-trivial isotrivial elliptic fibrations

Let $k$ be a number field. A non-trivial isotrivial elliptic fibration is an elliptic fibration over $\mathbb{P}^1$ that is isotrivial but not trivial. Density conjecture. The $k$-points are dense on any non-trivial isotrivial elliptic fibration over $\mathbb{P}^1$. This conjecture is relevant to the study of quadratic points on products of an elliptic curve and a genus $2$ curve. It is explicitly stated for $k=\mathbb{Q}$ in the cited work and is attributed there as implicit in earlier work.