Sustained simplicity of the nilpotent socle of simple Whittaker modules

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

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

Let $L$ be a simple Whittaker module for the Whittaker pair $(\mathfrak{g},\mathfrak{n}_+)$. Write $\operatorname{soc}_{\mathfrak{n}}(L)$ for its $\mathfrak{n}$-socle. The socle conjecture. The module $\operatorname{soc}_{\mathfrak{n}}(L)$ is simple. This is proposed for triangular decompositions in the sense of McDowell and is supported by examples, but the source also gives a counterexample in the general case without this hypothesis.