Harpaz–Wittenberg conjecture on specializations of norm equations

Let $P_1,\ldots,P_n\in\mathbb{Q}[t]$ be pairwise distinct irreducible monic polynomials. Let $k_i=\mathbb{Q}[t]/(P_i(t))$ be the corresponding number fields, and let $a_i\in k_i$ denote the class of $t$. For each $i$, let $L_i/k_i$ be a finite extension and let $b_i\in k_i^*$. Let $S_0$ be a finite set of places of $\mathbb{Q}$ containing the archimedean place and every finite place above which, for some $i$, either $b_i$ is not a unit or $L_i/k_i$ is ramified. For each $v\in S_0$, fix $t_v\in\mathbb{Q}_v$, and suppose that for every $i$ and $v$ there exists $x_{i,v}\in (L_i\otimes_{\mathbb{Q}}\mathbb{Q}_v)^*$ such that

$$b_i(t_v-a_i)=N_{L_i\otimes_{\mathbb{Q}}\mathbb{Q}_v/k_i\otimes_{\mathbb{Q}}\mathbb{Q}_v}(x_{i,v})$$

in $k_i\otimes_{\mathbb{Q}}\mathbb{Q}_v$. Harpaz–Wittenberg conjecture. There exists $t_0\in\mathbb{Q}$ arbitrarily close to $t_v$ for every $v\in S_0$, such that, for every $i$ and every place $\mathfrak{p}$ of $k_i$ with $\operatorname{ord}_{\mathfrak{p}}(t_0-a_i)>0$, either $\mathfrak{p}$ lies above a place in $S_0$ or $L_i$ possesses a place of degree $1$ over $\mathfrak{p}$. This conjecture predicts a strong specialization property for families of norm equations and is used in the study of the Hasse principle for varieties over number fields. The paper applies a sieve method to establish cases of the conjecture, while the general statement remains open.