Integral weak Hilbert property conjecture over arithmetic schemes

Let $S$ be a regular connected dominant $\mathbb{Z}$-variety, with function field $K=R(S)$. Let $\mathcal{Y}$ be a smooth $S$-variety and assume that $Y:=\mathcal{Y}_K$ is connected. Let $\mathcal{Y}(S)^{(1)}$ denote the set of near $S$-integral points of $Y$.

Integral weak Hilbert property conjecture. If $\mathcal{Y}(S)^{(1)}$ is Zariski dense in $Y$, then $\mathcal{Y}$ has WHP over $S$ potentially.

This conjecture is motivated by work of Corvaja and Zannier and agrees essentially with a conjecture of Luger. Its general status is not specified in the supplied text.