Stable-pairs rationality conjecture for vertical classes on K3 fibrations

Let $V$ be a nonsingular projective threefold that is the total space of a K3 fibration $\pi:V\to C$ over a nonsingular projective curve, and let $\beta\in H_2(V,\mathbb{Z})^\pi$ be a nonzero vertical class. Define

$${\mathsf{Z}}_{\mathsf{P}}(V;q)_\beta=\sum_n \widetilde{N}^{\bullet}_{n,\beta}q^n,$$

where $\widetilde{N}^{\bullet}_{n,\beta}$ is the stable-pairs invariant. Stable-pairs rationality conjecture. The partition function ${\mathsf{Z}}_{\mathsf{P}}(V;q)_\beta$ is the Laurent expansion of a rational function in $q$.

This is presented as a special case of the stable-pairs rationality conjecture for threefolds. The source notes that the partition function is a Laurent series because the stable-pairs moduli spaces are empty for sufficiently negative Euler characteristic.