Characterize the weight filtration via stabilized root multiplicities

Determine whether, for every simple regular holonomic $D_X$-module $S$, its localization $M=S_f\neq 0$ along a holomorphic function $f$, every complex number $\alpha$, and every nonnegative integer $\ell$, the weight-filtration piece satisfies \[ W_{q+\ell}(M\cdot f^{-\alpha})=D_X\cdot\{mf^{-\alpha}\mid \nu_{m,\alpha}\leq \ell\}, \] where $q$ is the weight of $S$ and $\nu_{m,\alpha}$ is the stabilized multiplicity of the root $s=-\alpha$ in the power $b$-functions of $m$ with respect to $f$.

Background

The paper defines the numerical invariant $\nu_{m,\alpha}$ from the eventual multiplicity of $s=-\alpha$ in the power Bernstein–Sato polynomials of an element $m$ of the localized $D$-module $M=S_f$. The associated sets $N_\ell=\{mf^{-\alpha}\mid \nu_{m,\alpha}\leq\ell\}$ are shown to be quasi-coherent $O_X$-submodules contained in the corresponding weight-filtration pieces $W_{q+\ell}(M\cdot f^{-\alpha})$.

The authors prove equality after applying the $D_X$-action in the lowest weight level and at the top level, and establish equality in several special settings, including isolated quasihomogeneous singularities, hyperplane arrangements, and spherical varieties. The unresolved issue is whether the $D_X$-submodule generated by each $N_\ell$ always equals the full weight-filtration piece.