Topological multivariable strong monodromy conjecture

Let $f=f_{1}\cdots f_{r}$ with each $f_{k}\in\mathbb{C}[x_{1},\dots,x_{n}]$, and set $F=(f_{1},\dots,f_{r})$. Let $\mu:Y\to X$ be a log resolution of $f$. Write $\{E_{i}\}_{i\in S}$ for the irreducible components of $f\circ\mu$, let $a_{i,j}$ be the order of vanishing of $f_{j}$ along $E_{i}$, let $k_{i}$ be the order of vanishing of the determinant of the Jacobian of $\mu$ along $E_{i}$, and for $I\subseteq S$ define

$$E_{I}^{\circ}=\bigcap_{i\in I}E_{i}\setminus\bigcup_{i\in S\setminus I}E_{i}.$$

The topological zeta function is

$$Z_{F}^{\operatorname{top}}(S)=\sum_{I\subseteq S}\chi(E_{I}^{\circ})\prod_{i\in I}\frac{1}{a_{i,1}s_{1}+\cdots+a_{i,r}s_{r}+k_{i}+1},$$

and $B_{F}$ is the Bernstein–Sato variety.

Topological multivariable strong monodromy conjecture. The polar locus of $Z_{F}^{\operatorname{top}}(S)$ is contained in $V(B_{F})$.

This conjecture relates poles of the resolution-theoretic topological zeta function to Bernstein–Sato varieties. The supplied text states the conjecture but gives no resolution evidence, so its status is recorded as open.