Simplicity conjecture for generic radical central characters

Let $\mathfrak{g}=\mathfrak{sl}_2\ltimes L(4)$, let $\chi$ be a radical central character, and let $Q(0,\chi)$ denote the corresponding module. Suppose that

$$\chi\neq(\mu^2,\mu^3)\quad\text{for every }\mu\in\mathbb{C}.$$

Generic-character simplicity conjecture. The module $Q(0,\chi)$ is simple.

The conjecture concerns the radical central characters not parametrized by $(\mu^2,\mu^3)$. The source explains that these cases cannot be obtained by localizing highest-weight modules and that a generalization of highest-weight theory would be needed to treat them; no resolution is given.