Motivic zeta-function rationality for curves with finite group actions

Let $G$ be a finite group of order $r$, let $C$ be a nonsingular projective curve over an algebraically closed field $k$ of characteristic $0$ or of positive characteristic $p$ with $p\nmid r$, and let $\sigma:G\times C\rightarrow C$ be a group action on $C$. Write

$$\zeta_{(C,\sigma)}(t)=\sum_{n=0}^{\infty}[\operatorname{Sym}^n(C,\sigma)]t^n.$$

Finite-group generalization conjecture. The motivic zeta function $\zeta_{(C,\sigma)}(t)$ is rational.

The paper proves this rationality when $G$ is finite abelian. The conjecture seeks the corresponding result for arbitrary finite groups; the obstacle is handling the group action on the relevant vector bundles without diagonalizing the representation.