Strong approximation conjecture for varieties associated to pairs of polynomials and field extensions

Let $k$ be a number field, let $P_1(t),\cdots, P_n(t)$ be pairwise distinct irreducible polynomials in $k[t]$. Set $k_i=k[t]/(P_i(t))$, let $a_i$ be the class of $t$ in $k_i$, let $L_i/k_i$ be finite field extensions, and let $b_i\in k_i^*$. Let $F_i$ be the singular locus of $R_{L_i/k}(\mathbb A^1_{L_i})\setminus R_{L_i/k}(\mathbb G_{m,L_i})$. Let $W$ be the closed subvariety of $\mathbb A^2_k\setminus \{(0,0)\}\times \prod_{i=1}^n (R_{L_i/k}(\mathbb A^1_{L_i})\setminus F_i)$ with coordinates $(\lambda,\mu,{\bf z}_1,\cdots,{\bf z}_n)$ defined by

$$b_i(\lambda-a_i\mu)=N_{L_i/k_i}({\bf z}_i),\qquad 1\leq i\leq n,$$

where $N_{L_i/k_i}$ is the norm form of $L_i/k_i$. Such a $W$ is a variety associated to the pairs $(P_1(t),L_1),\cdots,(P_n(t),L_n)$. Strong approximation conjecture. The variety $W$ satisfies strong approximation for integral points off any place $v_0$ of $k$. This conjecture concerns the fibration method and would imply Harpaz and Wittenberg's Conjecture 9.1; the assertion remains unresolved in the supplied source.