Conjecture F for fibrations with split fibres away from a finite locus

Let $k$ be a number field. Let $\pi\in{\mathscr P}$ determine a variety $W$ and a smooth morphism $p:W\to {\mathbf P}^1_k$, with split fibres above $U={\mathbf P}^1_k\setminus M$. For $c\in {\mathbf P}^1_k$, write $W_c=p^{-1}(c)$.

Conjecture F. The subset

$$\bigcup_{c\in U(k)}W_c({\mathbf A}_k)$$

is dense in $W({\mathbf A}_k)$.

This is an arithmetic conjecture concerning adelic points on fibrations over the projective line, equivalent to Conjecture 9.1 of the cited work. Its status is not specified in the source.