The Bernstein-Sato zero-locus and Sabbah specialization conjecture

Let $F=(f_1, 3ellots,f_r)$ be a collection of non-zero polynomials on $X=\mathbb{C}^n$, let $D=\bigcup_{j=1}^rV(f_j)$, and let $B_{F,x}$ denote the local Bernstein-Sato ideal at $x\in D$. Let

$$\operatorname{Exp}:\mathbb{C}^r\longrightarrow(\mathbb{C}^*)^r, 3\qquad x\longmapsto\exp(2\pi i x),$$

and let $\psi_F$ be the Sabbah specialization functor, with uniform support $\operatorname{Supp}^{\mathrm{unif}}_x(\psi_F\mathbb{C}_X)\subset(\mathbb{C}^*)^r$. Bernstein-Sato specialization conjecture.

$$\operatorname{Exp}(V(B_{F,x}))=\bigcup_{y\in D\text{ near }x}\operatorname{Supp}^{\mathrm{unif}}_y(\psi_F\mathbb{C}_X).$$

This is proposed as a multivariable generalization of the Kashiwara–Malgrange relation between Bernstein-Sato roots and Milnor-fiber monodromy eigenvalues. The statement connects algebraic Bernstein-Sato ideals with cohomology support loci of local systems, but its general validity remains open.