Holomorphy conjecture for the twisted topological zeta function

Let $f:X\to\bb A^1_{\bb C}$ be as in the construction of the twisted topological zeta function, let $e\geq 1$, and write $Z^{(e)}_{{\rm top}}(s)$ for that rational function. A monodromy eigenvalue has order divisible by $e$ when its multiplicative order is divisible by $e$. Holomorphy conjecture. The function $Z^{(e)}_{{\rm top}}(s)$ is a polynomial in $s$, unless there is an eigenvalue with order divisible by $e$ of the monodromy action on the cohomology of the Milnor fiber at some point of the locus of $f$. This conjecture predicts that poles of the twisted topological zeta function can occur only in the presence of the specified monodromy eigenvalue.