Elliptic-fibration quasimodularity conjecture

Let $X$ and $B$ be nonsingular projective varieties, let $cpi:X\to B$ be an elliptic fibration with integral fibers and a section $ciota:B\to X$, and let $N_{\iota}$ be the normal bundle of the section. Define

$$W=B_0-\frac{1}{2}\pi^*c_1(N_{\iota}),$$

where $B_0$ is the class of the section, and define the generating series ${\mathcal C}^{\pi}_{g,\mathsf{k}}$ from the relative Gromov--Witten classes for curve classes mapping to $\mathsf{k}\in H_2(B,\mathbb Z)$. Elliptic-fibration quasimodularity conjecture. For any $\gamma_1,\ldots,\gamma_n\in H^*(X)$ and $\mathsf{k}\in H_2(B,\mathbb Z)$,

$${\mathcal C}^{\pi}_{g,\mathsf{k}}(\gamma_1,\ldots,\gamma_n)\in H_*(\overline M_{g,n}(B,\mathsf{k}))\otimes\frac{1}{\Delta(q)^m}\mathsf{QMod},$$

where $m=-\frac{1}{2}c_1(N_{\iota})\cdot\mathsf{k}$. This predicts quasimodularity, with a controlled pole at the discriminant, for the relative Gromov--Witten generating series of elliptic fibrations. The paper notes that the conjecture is proved for the trivial elliptic fibration $X=B\times E$ after the relevant specialization.