The local monodromy conjecture for motivic and topological zeta functions

Let $f \in \mathbb{C}[x_1,\dots,x_n] \setminus \mathbb{C}$ and assume $f(o)=0$. The local motivic zeta function $Z_{\operatorname{mot},o}(f;s)$ and the local topological zeta function $Z_{\operatorname{top},o}(f;s)$ are rational functions associated with the germ of $f$ at $o$. A monodromy eigenvalue of $f$ at a point $p\in f^{-1}\{0\}$ is an eigenvalue of the local Milnor monodromy acting on some cohomology group of the Milnor fibre at $p$. Monodromy conjecture. If $s_0$ is a pole of $Z_{\operatorname{top},o}(f;s)$ or $Z_{\operatorname{mot},o}(f;s)$, then

$$e^{2\pi i s_0}$$

is a monodromy eigenvalue of $f$ at some point of $f^{-1}\{0\}$ close to $o$. The conjecture relates poles of singularity invariants to the topology of the Milnor fibre; the paper states that it provides counterexamples to the statement in the cases it studies, so this formulation is not established in general.