Central generation of Whittaker vectors in standard Whittaker modules

Let $\mathfrak{g}$ be a Lie algebra with a fixed triangular decomposition

$$\mathfrak{g}=\mathfrak{n}_-\oplus\mathfrak{h}\oplus\mathfrak{n}_+,$$

and set $\mathfrak{n}=\mathfrak{n}_+$. Let $\lambda\in(\mathfrak{n}/[\mathfrak{n},\mathfrak{n}])^*$, let $L_\lambda$ be the one-dimensional $\mathfrak{n}$-module defined by $\lambda$, choose a basis vector $v_\lambda$, and set $M_\lambda=U(\mathfrak{g})\otimes_{U(\mathfrak{n})}L_\lambda$. Central-generation conjecture. For generic $\lambda$, the center $Z(\mathfrak{g})$ of $U(\mathfrak{g})$ surjects onto the set of Whittaker vectors of $M_\lambda$ via

$$z\longmapsto z\otimes v_\lambda,\qquad z\in Z(\mathfrak{g}).$$

This predicts that, generically, every Whittaker vector in the standard module arises from the center. The source presents it as a conjecture based on examples.