Duality involution conjecture for Whittaker modules

Let $\mathfrak{g}$ be the Lie algebra under consideration, let $\eta$ be a character defining the category $\mathcal{N}_\eta$ of Whittaker modules, let $W_\eta$ be the corresponding Weyl-group subgroup, and let $X_\mu$ denote the generalized $\mathfrak{z}$-weight spaces of $X$ as in the definition of $X^{\vee_\beta}$. For nonzero $\beta\in\mathbb{C}[W_\eta]$, set

$$X^{\vee_\beta}:=\bigoplus_\mu X_\mu^{\vee_\beta}.$$

Duality involution conjecture. If $X\in\mathcal{N}_\eta$ and $\beta\in\mathbb{C}[W_\eta]$ is nonzero, then $X^{\vee_\beta}\in\mathcal{N}_\eta$ and

$$(X^{\vee_\beta})^{\vee_\beta}\cong X.$$

If $\eta=0$, then $X^{\vee_1}=X^\vee$ is the usual category $\mathcal{O}$ duality functor. This would extend the familiar category $\mathcal{O}$ duality to Whittaker modules, but the asserted closure and involutivity for general nonzero $\beta$ remain unproved in the source.