The Bernstein-Sato specialization equality conjecture

Let $F=(f_1, 3ellots,f_r)$, let $M\in\mathbb{N}^{p\times r}$ be non-zero, let ${\bf m}\in\mathbb{N}^p$, and let $F^M$ and the monomial maps $\phi_M$ be defined as in the source. Write $B_F^{{\bf m}\cdot M}$ and $B_{F^M}^{\bf m}$ for the corresponding Bernstein-Sato ideals. Bernstein-Sato specialization equality conjecture. For all non-zero $M\in\mathbb{N}^{p\times r}$ and all vectors ${\bf m}\in\mathbb{N}^p$,

$$\phi_M^{-1}(\operatorname{Exp}(V(B_F^{{\bf m}\cdot M})))=\operatorname{Exp}(V(B_{F^M}^{\bf m})).$$

In particular, for all $M$ with nonzero columns,

$$\phi_M^{-1}(\operatorname{Exp}(V(B_F)))=\operatorname{Exp}(V(B_{F^M})).$$

The preceding proposition establishes only an inclusion, so this proposed converse equality is not proved there. Its general validity is open.