Budur's multivariable monodromy conjecture

Let ${\bf f}=(f_1,f_2,\dots,f_r)$ be an $r$-tuple of polynomials in $\mathbb{C}[x_1,x_2,\dots,x_n]$, and let $D_i=(f_i=0)$ and $D=\sum_{i=1}^rD_i$. For a log resolution $\mu\colon (Y,F)\to (X,D)$, where $X=\mathbb{C}^n$ and $F$ has irreducible components $E_i$ indexed by $i\in S$, write

$$\mu^*D_j=\sum_i a_{ij}E_i,\qquad K_{Y/X}=\sum_i k_iE_i.$$

Define the topological zeta function and its pole locus by

$$Z_{\bf f}({\bf s})=\sum_{I\subseteq S}\chi(E_I^o)\prod_{i\in I}\frac{1}{\sum_{j=1}^r a_{ij}s_j+k_i+1},$$

where $E_I^o=\bigcap_{i\in I}E_i\setminus\bigcup_{i\in S\setminus I}E_i$, and let $\textup{PL}(Z_{\bf f}({\bf s}))$ be the pole locus of $Z_{\bf f}({\bf s})$. Let $B_{\bf f}$ denote the multivariable Bernstein–Sato ideal, and write $Z(B_{\bf f})$ for its zero locus.

Budur's multivariable monodromy conjecture.

$$\textup{PL}(Z_{\bf f}({\bf s}))\subseteq Z(B_{\bf f}).$$

This is a multivariable analogue of the monodromy conjecture, relating poles of the topological zeta function to the zero locus of the Bernstein–Sato ideal. The source attributes this formulation to Budur's Conjecture 1.17 and gives no resolution status.