Generic uniqueness and annihilator conjecture for Whittaker modules

Let $\mathfrak{g}$ be in the setup of the preceding construction, let $\mu\in\mathfrak{h}^*$ be such that the Verma module $M(\mu)$ is simple, and let $\lambda\in(\mathfrak{n}/[\mathfrak{n},\mathfrak{n}])^*$. Let $L(\lambda,\mu)$ denote the simple Whittaker module constructed from these parameters. Generic uniqueness conjecture. For generic $\lambda$, $L(\lambda,\mu)$ is the unique, up to isomorphism, simple Whittaker module in $\mathfrak{W}_{\mathfrak{n}}^{\mathfrak{g}}(\lambda)$ whose annihilator coincides with $\operatorname{Ann}_{U(\mathfrak{g})}L(\lambda,\mu)$. The claim refines the preceding existence and annihilator results, but the source supplies no resolution.