Finiteness conjecture for Picard rank jumps in Lefschetz pencils

Let $\mathcal{Y}\to\mathbb{P}^1$ be a Lefschetz pencil of degree-$d$ surfaces in $\mathbb{P}^3$, and let $\mathcal{Y}_s$ denote its fiber over $s\in\mathbb{P}^1(\mathbb{C})$. Assume $d\geq 5$.

Picard-rank finiteness conjecture. The Picard number of $\mathcal{Y}_s$ is greater than or equal to $2$ for at most finitely many values of $s\in\mathbb{P}^1(\mathbb{C})$.

This is a special conjectural case involving zero-dimensional components and is described as beyond the authors' current understanding. It concerns finiteness of fibers with an extra Picard class in a Lefschetz pencil.