Budur's linear-factor generation conjecture for Bernstein-Sato ideals

Let $F=f_1,\dots,f_\ell$ be a tuple of elements of $\mathbb{C}\{x_1,\dots,x_d\}$, and let $B_F\subseteq\mathbb{C}[s_1,\dots,s_\ell]$ be its Bernstein–Sato ideal. A linear polynomial has the form

$$a_1s_1+\cdots+a_\ell s_\ell+a,$$

with $a_i\in\mathbb{Q}_{\geq 0}$ and $a\in\mathbb{Q}_{>0}$. Budur's conjecture. The ideal $B_F$ is generated by products of such linear polynomials. The conjecture seeks a multivariable analogue of the factorization of Bernstein–Sato polynomials into linear factors. The supplied source explains that it was prompted by rationality results of Gyoja and Sabbah, but gives no resolution status.