Tanisaki's conjecture on characters of primitive ideal quotients

Let $\mathfrak{g}$ be a simple Lie algebra, let $U(\mathfrak{g})$ be its universal enveloping algebra, and let $\mathscr{I}_0$ be the set of primitive ideals with trivial central character. For $w\in W$, write $I_w=\operatorname{Ann}(L_w)$. Tanisaki's conjecture. If $I_w,I_y\in\mathscr{I}_0$ and the characteristic varieties of the corresponding primitive ideal quotients agree, then

$$\operatorname{Ch}(U(\mathfrak{g})/I_w)=\operatorname{Ch}(U(\mathfrak{g})/I_y)$$

implies $I_w=I_y$, equivalently $w\stackrel{L}{\sim}y$. The statement concerns whether the characteristic variety determines a primitive ideal among those with trivial central character. The supplied text does not give evidence that this conjecture has been resolved.