Budur's hyperplane-arrangement Bernstein–Sato variety conjecture

Let $\mathcal{A}$ be a central, essential, indecomposable hyperplane arrangement in $\mathbb{C}^{n}$, with defining polynomial factored as

$$f_{\mathcal{A}}=f_{1}\cdots f_{r}.$$

Assume each $f_{k}$ is a (not necessarily reduced) central hyperplane arrangement of degree $d_{k}$, and set $F=(f_{1},\dots,f_{r})$. Let $B_{F}$ denote the Bernstein–Sato variety of $F$.

Budur's conjecture. The affine hyperplane

$$\{(s_{1},\dots,s_{r})\in\mathbb{C}^{r}\mid d_{1}s_{1}+\dots+d_{r}s_{r}+n=0\}$$

is contained in $V(B_{F})$.

This conjecture generalizes the one-variable $-n/d$ conjecture for central, essential, indecomposable hyperplane arrangements. The paper proves the assertion when the arrangement is tame; the general case remains open in the source.