Abelian-surface multiple-cover conjecture

Let $A$ be an abelian surface, let $\beta\in H_2(A,\mathbb{Z})$ be effective, and for each divisor $k\mid\beta$ choose an abelian surface $A_k$ and a parallel-transport morphism $\varphi_k$ taking $\beta/k$ to a primitive effective class. Let $\alpha\in H^{\ast}(\overline M_{g,n})$ be tautological and let $\gamma_1,\ldots,\gamma_n\in H^{\ast}(A,\mathbb{R})$. Abelian-surface multiple-cover conjecture.

$$\left\langle\alpha;\gamma_1,\ldots,\gamma_n\right\rangle^A_{g,\beta}=\sum_{k\mid\beta}k^{3g-3+n-\deg(\alpha)}\left\langle\alpha;\varphi_k(\gamma_1),\ldots,\varphi_k(\gamma_n)\right\rangle^{A_k}_{g,\varphi_k(\beta/k)}.$$

Equivalently, after subtracting multiple covers, the reduced Gromov–Witten invariants are independent of divisibility. The rule extends a proposal for the abelian-surface analogue of the Katz–Klemm–Vafa formula and remains open.