One-singularity curve auto-Igusa zeta structure conjecture

Let $C$ be a connected curve over $k$ with only one singular point $p$, let $f:\bar C\to C$ be its normalization, and let $\bar\zeta_{C,p}(t)$ be the reduced auto-Igusa zeta series. Let $\Theta_{W^{\deg(f)},\mathfrak{l}}^{\star_q}(t)$ denote the indicated restricted motivic Igusa series, where $W$ is a connected curve analytically isomorphic to $C$ at $O$. One-singularity curve auto-Igusa zeta conjecture. There exist $r,b,q\in\mathbb{N}$ such that

$$\bar\zeta_{C,p}(t^r)=\frac{1-t^{r(b-1)}}{1-t^r}+t^{rb}\Theta_{W^{\deg(f)},\mathfrak{l}}^{\star_q}(t)$$

as elements of $\mathcal{G}_k[[t]]$. The formula is proposed after explicit rational calculations for smooth curves, the cusp, the node, and the nodal cubic; its general validity for one-singularity curves remains open.