The holomorphy conjecture for ideals

Let $\mathcal{I}$ be an ideal defining a subscheme, let $\pi$ be an embedded resolution of $\mathcal{I}$, and let $Z_{\mathrm{top},\mathcal{I}}^{(d)}$ denote the associated topological zeta function of order $d$. Let $d$ be a positive integer. Holomorphy conjecture. If $d$ does not divide the order of any eigenvalue of monodromy associated to the ideal $\mathcal{I}$ at points of $\pi^{-1}\{0\}$, then $Z_{\mathrm{top},\mathcal{I}}^{(d)}$ is holomorphic on the complex plane. The conjecture extends the holomorphy phenomenon from hypersurfaces to arbitrary ideals; the paper proves it when $\mathcal{I}$ is generated by finitely many complex polynomials in two variables, while the general case remains open.