Geometric Budur-type conjecture for Bernstein–Sato ideals

Let $X$ be the underlying space, let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_X$-module, and let $\mathcal{N}_0$ be the associated module. For a monoid ideal $K\subseteq\mathbb{N}^r$, let $K_{\mathbf{m}}$ denote the construction associated with $K$ and $\mathbf{m}\in\mathbb{N}^r$, and let $Z(B^{K_{\mathbf{m}}}(\mathcal{N}_0))$ be the zero locus of the corresponding Bernstein–Sato ideal. Geometric Budur-type conjecture.

$$Z(B^{K_{\mathbf{m}}}(\mathcal{N}_0))\subseteq\mathbb{C}^r$$

is a finite union of translated linear subvarieties. Moreover, if $\mathcal{M}$ underlies a $\mathbb{Q}$-mixed Hodge module, then

$$Z(B^{K_{\mathbf{m}}}(\mathcal{N}_0))\subseteq\mathbb{C}^r$$

is a finite union of $\mathbb{Q}$-translated linear subvarieties. The paper presents the containment of the zero locus in a union of such subvarieties as geometric evidence for Budur's conjecture; the conjecture asks for the stronger finite-union statement in this general setting.