Bernstein–Sato support conjecture for Sabbah's specialization complex

Let $F=(f_1,\ldots,f_r):X\rightarrow\mathbb{C}^r$ be a collection of non-invertible analytic functions on a complex manifold such that $f^{-1}(0)$ admits a finite Whitney stratification, where $f=\prod_{i=1}^r f_i$. Define $\mathcal{S}(F)$ as the support of Sabbah's specialization complex, and let $\mathcal{Z}(B_F)$ be the zero locus of the Bernstein–Sato ideal of $F$. Let

$$\operatorname{Exp}:\mathbb{C}^r\rightarrow(\mathbb{C}^*)^r,$$

be given by $\alpha\mapsto\exp(2\pi i\alpha)$. Bernstein–Sato support conjecture. One has

$$\mathcal{S}(F)=\operatorname{Exp}(\mathcal{Z}(B_F)).$$

This conjecture gives a Bernstein–Sato-ideal description of the support of Sabbah's specialization complex, avoiding a log resolution. The source indicates that one inclusion, $\mathcal{S}(F)\subset\operatorname{Exp}(\mathcal{Z}(B_F))$, has been proved; the equality is therefore solved.