Fibration conjecture for the weak Hilbert property

Let $K$ be a finitely generated field of characteristic zero. Let $Y$ and $Z$ be smooth connected $K$-varieties, and let $f:Y\to Z$ be a surjective morphism. Assume that $Z$ has weak Hilbert property (WHP) over $K$. For $z\in Z(K)$, write $Y_z=Y\times_{Z,f}\operatorname{Spec}(k(z))$.

Fibration conjecture for WHP. (a) Let $\Xi$ be the set of all $z\in Z(K)$ such that $Y_z$ has WHP over $K$. If $\Xi$ is not strongly thin in $Z$, then $Y$ has WHP over $K$. (b) If the generic fibre of $f$ has WHP over the function field $R(Z)$ of $Z$, then $Y$ has WHP over $K$.

This is a fibration principle for transferring the weak Hilbert property from the base and suitable fibres to the total space. It is open, although known in some cases.