The K3 multiple cover formula for reduced Gromov–Witten invariants

Let $S$ be a K3 surface, let $\beta$ be an effective curve class, and for every divisor $k\mid\beta$ let $S_k$ be a K3 surface with a complex isometry $\varphi_k$ such that $\varphi_k(\beta/k)$ is primitive and effective. Extend $\varphi_k$ to the full cohomology by $\varphi_k(1)=1$ and $\varphi_k(\mathsf{p})=\mathsf{p}$. Let $\gamma_1,\ldots,\gamma_r$ be insertions and let $\tau_{k_i}$ denote descendent insertions.

Multiple Cover Formula.

$$\left\langle\tau_{k_1}(\gamma_1)\cdots\tau_{k_r}(\gamma_r)\right\rangle^{S,\mathsf{GW},\mathsf{red}}_{g,\beta}=\sum_{k\mid\beta}k^{2g-3+\sum_i\deg(\gamma_i)}\left\langle\tau_{k_1}(\varphi_k(\gamma_1))\cdots\tau_{k_r}(\varphi_k(\gamma_r))\right\rangle^{S,\mathsf{GW},\mathsf{red}}_{g,\varphi_r(\beta/k)}.$$

This formula was conjectured in the cited work on K3 surfaces and expresses imprimitive reduced Gromov–Witten invariants as a divisor sum of primitive invariants. Its status is not resolved by the supplied material.