The generating-level bound for the Hodge filtration of a free divisor

Let $X$ be a complex manifold, let $D$ be a divisor satisfying the assumptions of Theorem and Corollary, and let $b_D(s)$ be the relevant Bernstein–Sato polynomial. For a well-filtered module $(\mathcal{M},F_\bullet)$, the filtration is generated at level $k$ when

$$F_l\mathcal{D}_X\cdot F_k\mathcal{M}=F_{k+l}\mathcal{M}$$

for every $l\geq 0$; its generating level is the smallest such integer. Generating-level conjecture. The Hodge filtration $F^H_\bullet\mathcal{O}_X(*D)$ is generated at level

$$r:=\frac{1}{2}\left(\deg(b_D(s))-\operatorname{mult}_{b_D(s)}(-1)\right).$$

This conjecture gives a bound for the complexity of the Hodge filtration and is supported by the calculations in the paper. Its general validity under the stated assumptions is not established here.