Tanisaki's conjecture on associated varieties of orbital varieties

Let $\mathcal{O}$ be a special nilpotent orbit, let $\mathscr{C}$ be the corresponding double cell, and let $\mathscr{C}/ {\small \stackrel{R}{\sim}}$ be the set of right cells contained in $\mathscr{C}$. Write $\operatorname{Irr}(\overline{\mathcal{O}}\cap \mathfrak{n})$ for the irreducible components of $\overline{\mathcal{O}}\cap \mathfrak{n}$. Tanisaki's conjecture. There exists a bijection $\varphi$ from $\mathscr{C}/ {\small \stackrel{R}{\sim}}$ to $\operatorname{Irr}(\overline{\mathcal{O}}\cap \mathfrak{n})$, with $\varphi(w)=Y_w$, and an ordering $\prec$ on this set such that

$$V(L_w)=Y_w\cup\widetilde{Y}_w.$$

Here $\widetilde{Y}_w$ is a union of orbital varieties $\mathcal{V}(y)$ satisfying $\mathcal{V}(y)\prec Y_w$. The conjecture is used in the paper as part of the study of associated varieties and primitive ideals; the supplied text does not establish its resolution, so its status remains open here.