Recover the Hodge filtration from the p-function filtration

Determine whether the recursively defined filtration \[ Q_{\ell}(M\cdot f^{-\alpha})=P_{\ell}(M\cdot f^{-\alpha})+F_1D_X\cdot Q_{\ell-1}(M\cdot f^{-\alpha}) \] coincides with the Hodge filtration $F_{\ell}(M\cdot f^{-\alpha})$ for $M=(O_X)_f$ and rational $\alpha$, where $P_\ell$ is the filtration defined by the degrees of the $p$-functions.

Background

For $S=O_X$ and $M=(O_X)_f$, the paper defines $p$-functions $p_{m,\alpha}(s)$ and proves that $mf^{-\alpha}$ belongs to the Hodge-filtration level indexed by $\deg p_{m,\alpha}(s)$. This yields a $P$-filtration contained in the Hodge filtration, but the $P$-filtration need not satisfy Griffiths transversality.

To correct this defect, the authors enlarge $P_\ell$ recursively using the action of first-order differential operators and define $Q_\ell$. They leave unresolved whether this corrected filtration exactly recovers the Hodge filtration in general, although equality is established in the spherical-variety setting.