Noguchi's conjecture on the zeta function and Euler characteristic of finite categories

Let $I$ be a finite category with object set $\{x_1,\dots,x_n\}$. Let $A_I=(\#\operatorname{Hom}(x_i,x_j))_{i,j}$ be its adjacency matrix, and suppose that its series Euler characteristic $\chi_{\Sigma}(I)$ exists. Let

$$\zeta_I(z)=\exp\left(\sum_{m=1}^{\infty}\frac{\#N_m(I)}{m}z^m\right)$$

be the zeta function of $I$.

Noguchi's conjecture. The zeta function is a finite product of the form

$$\zeta_I(z)=\prod\frac{1}{(1-\alpha_i z)^{\beta_i}}\exp\left(\sum\frac{\gamma_jz^j}{j(1-\delta_jz)^j}\right),$$

for some complex numbers $\alpha_i,\beta_i,\gamma_j,\delta_j$, and the following assertions hold: $\sum\beta_i$ is the number of objects of $I$; each $\alpha_i$ is an eigenvalue of $A_I$, hence an algebraic integer; and

$$\sum\frac{\beta_i}{\alpha_i}+\sum(-1)^j\frac{\gamma_j}{\delta_j^{j+1}}=\chi_{\Sigma}(I).$$

The conjecture proposes a precise relationship between the zeta function and the series Euler characteristic of a finite category. The abstract states that it is verified for finite groupoids, finite acyclic categories, categories with two objects, and finite categories satisfying a certain condition; the general case remains open in the source.