Harari–Wittenberg's strong approximation conjecture for the fibration varieties $W$

Let $m_1,\dots,m_n$ be closed points of ${\mathbf A}^1_k$. For each $i$, let $L_i$ be a finite extension of the residue field $k_i=k(m_i)$, let $a_i\in k_i$ be the coordinate of $m_i$, and choose $b_i\in k_i^*$. Let $F_i$ be the singular locus of the affine variety defined by $N_{L_i/k}(x_i)=0$. For coordinates $\lambda,\mu$ on ${\mathbf A}^2_k$, let $W$ be the closed subvariety

$$W\subset\left({\mathbf A}^2_k\setminus\{(0,0)\}\right)\times\prod_{i=1}^n\left(R_{L_i/k}({\mathbf A}^1_{L_i})\setminus F_i\right)$$

defined by $N_{L_i/k_i}(x_i)=b_i(\lambda-a_i\mu)$ for all $i$, with the smooth surjective map $p:W\to{\mathbf P}^1_k$ given by $p(\lambda,\mu,x_1,\dots,x_n)=[\lambda:\mu]$. Write $W_c=p^{-1}(c)$.

Harari–Wittenberg's conjecture. The set

$$\bigcup_{c\in{\mathbf P}^1(k)}W_c({\mathbf A}_k)$$

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

This is a strong-approximation-type fibration conjecture used in the source to obtain positive results for fibrations over ${\mathbf P}^1$. The source gives no resolution.