The arithmetic weak Hilbert property conjecture

Let $R$ be an arithmetic ring with fraction field $k$, and let $\mathcal{X}$ be a regular quasi-projective arithmetic scheme over $R$. Write $\mathcal{X}(R)^{(1)}$ for the relevant set of integral points, and let $\mathcal{X}_k$ denote the generic fibre. Arithmetic weak Hilbert property conjecture. If $\mathcal{X}(R)^{(1)}$ is dense in $\mathcal{X}_k$, then there is a generically finite extension $S/R$ such that $\mathcal{X}_S$ has the weak Hilbert property over $S$. This is proposed as a natural extension of Corvaja–Zannier's conjectures to quasi-projective arithmetic schemes. The source records several classes of examples for which it is proved, including smooth affine tori over rings of $S$-integers and further varieties, but does not establish the conjecture in general.