Quasimodularity conjecture for quasimap potentials of elliptic fibrations

Let $X\rightarrow B$ be an elliptic fibration and, for $\gamma\in H_2(B,\mathbb Z)$, let $\mathcal{F}^{\mathsf{SQ}}_{g,\gamma}$ be the associated quasimap series. Define

$$L(q)=(1-27q)^{-1/3},\qquad B_1'(q)=q\frac{\partial}{\partial q}I_1^E(q),$$

and

$$X(q)=\frac{q\frac{\partial}{\partial q}B_1'(q)}{1+B_1'(q)},\qquad \mathsf{QEF}=\mathbb C[L^{\pm3},B_1',X].$$

Quasimap elliptic-fibration conjecture. For $\gamma\in H_2(B,\mathbb Z)$,

$$\mathcal{F}^{\mathsf{SQ}}_{g,\gamma}\in\mathsf{QEF}.$$

The conjecture is motivated by the expected relationship between quasimap and Gromov–Witten theories and by the B-model description; the source gives examples as evidence but does not specify a resolution.