Weak Hilbert property conjecture for geometric fibrations

Let $K$ be a field of characteristic zero, and let $Y$ and $Z$ be connected smooth $K$-varieties. Let $f:Y\to Z$ be a dominant morphism such that the generic fibre $Y_{R(Z)}$ is a smooth and proper $R(Z)$-variety satisfying WHP over $R(Z)$. Assume that $Z$ has HP over $K$.

Weak Hilbert property conjecture for geometric fibrations. In the situation above, $Y$ has WHP over $K$.

The source presents this as an open special case implied by the fibration conjecture when $K/\mathbb{Q}$ is finitely generated, while the corresponding smooth proper case is proved in the paper.