The holomorphy conjecture for twisted topological zeta functions

Let $f\in \mathbb C[x_{1},\dots,x_{n}]\setminus \mathbb C$ be a non-constant polynomial, let $d\in\mathbb Z_{\geq 1}$, and let $Z^{\mathrm{top},(d)}_{(f)}(s)$ denote the twisted topological zeta function of the principal ideal $(f)$ associated to $d$. Holomorphy conjecture. The function $Z^{\mathrm{top},(d)}_{(f)}(s)$ is a polynomial unless there exists an eigenvalue of the monodromy action whose order is divisible by $d$. This conjecture connects the holomorphy of twisted topological zeta functions with the orders of monodromy eigenvalues. The paper proves the conjecture for the determinantal varieties under consideration.