The Bernstein-Sato ideal factorization conjecture

Let $F=(f_1,\ldots,f_r)$ be a collection of non-zero polynomials $f_j$ in $\mathbb{C}[x_1,\ldots,x_n]$, and let $B_F\subset\mathbb{C}[s_1,\ldots,s_r]$ be its Bernstein-Sato ideal, generated by the polynomials $b$ for which

$$b(s_1, 3ellots,s_r)f_1^{s_1} 3ellots f_r^{s_r}=Pf_1^{s_1+1} 3ellots f_r^{s_r+1}$$

for an algebraic differential operator $P$. Bernstein-Sato ideal factorization conjecture. The ideal $B_F$ is generated by products of linear polynomials of the form

$$\alpha_1s_1+ 3ellots+\alpha_rs_r+\alpha$$

with $\alpha_j\in\mathbb{Q}_{\geq0}$ and $\alpha\in\mathbb{Q}_{>0}$. This would in particular imply the analogous statement for the radical ideal of $B_F$. The conjecture refines the result of Sabbah and Gyoja that $B_F$ contains at least one element of this form; for $r=1$ it is due to Kashiwara. Its proposed shape is supported by computations, while the general statement remains open.