Length of $D_Xf^{-\alpha}$ in the isolated singularity case

Let $f$ be a convergent power series of $n$ variables having an isolated singularity at 0. Set $(X,0)=({\bf C}^n,0)$. For $\alpha\in{\bf Q}$, we show that the length of the $D_X$-module $D_Xf^{-\alpha}$ coincides with $\widetilde{\nu}_{\alpha}+r_f\widetilde{\delta}_{\alpha}+1$. Here $\widetilde{\nu}_{\alpha}$ is the dimension of the graded piece ${\rm Gr}_V^{\alpha}$ of the $V$-filtration on the saturation of the Brieskorn lattice modulo the image of $N:=\partial_tt-\alpha$ on ${\rm Gr}_V^{\alpha}$ of the Gauss-Manin system, $r_f$ is the number of local irreducible components of $f^{-1}(0)$ (where $r_f=1$ if $n>2$), and $\widetilde{\delta}_{\alpha}:=1$ if $\alpha\in{\bf Z}_{>0}$, and 0 otherwise. This generalizes an assertion by T. Bitoun and T. Schedler in the weighted homogeneous case, where the saturation coincides with the Brieskorn lattice and $N=0$. In the semi-weighted-homogeneous case, the above formula implies certain sufficient conditions for their conjecture about the length of $D_Xf^{-1}$ to hold or to fail.