Zilber–Pink conjecture for Picard numbers in Lefschetz pencils

Let $\mathcal{Y}\to\mathbb{P}^1$ be a Lefschetz pencil of degree-$d$ surfaces in $\mathbb{P}^3$. Picard-number finiteness conjecture. If $d\geq 5$, the Picard number of $\mathcal{Y}_s$ is at least $2$ for at most finitely many values of $s\in\mathbb{P}^1(\mathbb{C})$. This is an analogue of the preceding unlikely-intersection prediction for the Noether–Lefschetz locus; the supplied text gives no resolution status.