Finite-generation conjecture for twisted quasimap series of ${\mathbb P}^2 \times {\mathbb P}^1$

Let

$$\mathcal{F}^{\mathsf{SQ}}_g(q_1,q_2)=\sum_{d_1,d_2\ge0}N^{\mathsf{SQ}}_{g,(d_1,d_2)}q_1^{d_1}q_2^{d_2}$$

be the $((3;2),(0;0))$-twisted genus-$g$ quasimap series of ${\mathbb P}^2\times{\mathbb P}^1$, and let

$$L(q_1)=(1-27q_1)^{-1/3},\qquad B_1'(q_1)=q_1\frac{\partial}{\partial q_1}I_1^E(q_1),$$

with $X(q_1)=q_1(\partial B_1'/\partial q_1)/(1+B_1')$. Twisted elliptic-fibration quasimap conjecture. One has

$$q_1\frac{\partial}{\partial q_1}\mathcal{F}^{\mathsf{SQ}}_1(q_1,q_2)\in\mathbb C[[q_2]][L^3,B_1',X],$$

and, for $g\ge2$,

$$\mathcal{F}^{\mathsf{SQ}}_g(q_1,q_2)\in\mathbb C[[q_2]][L^3,B_1',X].$$

The claim is based on low-degree calculations; the source does not state whether it has been resolved.