Whittaker-category equivalence for finite W-algebras

Let $ab$ be a $W_0$-orbit in $ab^*$, and let $M(\Lambda,\chi)$ and $L(\Lambda,\chi)$ be the standard and irreducible objects in the category $\mathcal O(\chi)$, while $M(\Lambda,e)$ and $L(\Lambda,e)$ are the corresponding objects in $\mathcal O(e)$. Let $Q_{\mathfrak l}$ be the $(U(\mathfrak g),U(\mathfrak g,e))$-bimodule appearing in the annihilator comparison. Whittaker-category equivalence conjecture. There should be an equivalence of categories

$$\mathbb W:\mathcal O(\chi)\longrightarrow\mathcal O(e)$$

such that

$$\mathbb W M(\Lambda,\chi)\cong M(\Lambda,e),\qquad \mathbb W L(\Lambda,\chi)\cong L(\Lambda,e)$$

for every $ab\in\mathcal L$. Moreover, $ab$ should respect annihilators in the sense that

$$\operatorname{Ann}_{U(\mathfrak g)}(M)=\operatorname{Ann}_{U(\mathfrak g)}\left(Q_{\mathfrak l}\otimes_{U(\mathfrak g,e)}\mathbb W M\right)$$

for every $M\in\mathcal O(\chi)$. This would extend the relationship between the Whittaker category and the category of modules for the finite W-algebra, while also preserving annihilator ideals; the supplied text presents it as a stronger conjecture and gives no resolution.