Noguchi's zeta-function conjecture for finite categories

Let $I$ be a finite category with series Euler characteristic. Write $A_I$ for its adjacency matrix and $\chi_{\Sigma}(I)$ for its series Euler characteristic. The zeta function of $I$ is

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

where $N_m(I)$ is the set of composable sequences of $m$ morphisms in $I$. Noguchi's conjecture. There exist complex numbers $\alpha_i,\beta_i,\gamma_j,\delta_j$ such that

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

and the following hold: $\sum\beta_i$ is the number of objects of $I$; every $\alpha_i$ 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 paper proves that this conjecture holds under the stated hypothesis, so its conjectural status is resolved. Related verification under additional conditions is attributed to Noguchi's earlier work.