Branching-number factorization conjecture for Cohen–Lenstra series of curve singularities

Let $p$ be an $\mathbb F_q$-point of a reduced curve $X$. Write $\widehat Z_{X,p}(x)$ for the associated Cohen–Lenstra series, and let $(xq^{-1};q^{-1})_\infty$ denote the $q$-Pochhammer product. Then $\widehat Z_{X,p}(x)$ has a meromorphic continuation to all of $\mathbb C$ and factors as

$$\widehat Z_{X,p}(x)=\frac{1}{(xq^{-1};q^{-1})_\infty^r}H_{X,p}(x),$$

where $r$ is the branching number of $X$ at $p$ and $H_{X,p}(x)$ is an entire power series. This conjecture proposes a uniform analytic description of the Cohen–Lenstra series for reduced curve singularities, extending the known smooth-point and nodal-singularity cases; its validity for general curve singularities remains open.