The topological monodromy conjecture for complex plane curve singularities

Let $h:Y\to (X,X_0)$ be an embedded resolution of the hypersurface $X_0=\{f=0\}$, with numerical data $N_i$ and $\nu_i$ defined by

$$h^{-1}(X_0)=\sum_{i\in S}N_iE_i,\qquad K_Y=h^*K_X+\sum_{i\in S}(\nu_i-1)E_i.$$

The associated topological zeta function is

$$Z_f^{\mathrm{top}}(s)=\sum_{I\subseteq S}\chi(E_I^{\circ})\prod_{i\in I}\frac{1}{N_is+\nu_i}.$$

Topological monodromy conjecture. If $\theta$ is a pole of $Z_f^{\mathrm{top}}(s)$, then $\exp(2\pi i\theta)$ is an eigenvalue of the monodromy $M_x^{(q)}$ for some $q\in\mathbb N$ and some closed point $x$ in $X_0$. This conjecture proposes a bridge between poles of the resolution-theoretic topological zeta function and monodromy eigenvalues; the supplied text gives no resolution status, so it is recorded as open.