The characteristic-cycle formula for parabolic Verma modules

Let $G$ be the reductive group in the construction, with Borel subgroup $B$ and parabolic subgroup $P$, and let $\mathscr W$ be its Weyl group. For $w\in\mathscr W$, let $w^{\max}$ denote the associated maximal representative, let $w_0$ be the longest Weyl-group element, and let $\mathfrak M_P(w^{\max}w_0\cdot0)$ be the $D$-module on $G/B$ associated with the parabolic Verma module $M_P(w^{\max}w_0\cdot0)$. Write $[\overline X_w]$ for the characteristic-cycle class of the corresponding irreducible component. Characteristic-cycle conjecture. For every $w\in\mathscr W$,

$$[\overline X_w]=[\operatorname{Ch}(\mathfrak M_P(w^{\max}w_0\cdot0))].$$

The source places this assertion in the characteristic-cycle analysis; the immediately preceding analogous proposition is stated for a regular central character and is proved later in the paper, while no resolution is supplied for this candidate.