Injectivity conjecture for the Whittaker pairing map

Assume that $\eta$ is a nondegenerate character, let $W$ be the Weyl group, and let $\mathcal{Z}_\mathfrak{g}$ be the category of modules used in the Whittaker induction functor $I_\eta$. For $U\in\mathcal{Z}_\mathfrak{g}$, let $\Phi^U_\beta:I_\eta(U^*)\to(I_\eta(U))^*$ be the morphism defined from the Whittaker pairing parameter $\beta\in\mathbb{C}[W]$. Injectivity conjecture. For every nonzero $\beta\in\mathbb{C}[W]$ and every $U\in\mathcal{Z}_\mathfrak{g}$, $\Phi^U_\beta$ is an injective morphism of $U(\mathfrak{g})$-modules. This is a proposed algebraic nondegeneracy property of the pairing between induced Whittaker modules and their duals; no proof or resolution is supplied in the source.