The extension of the codimension-one ideal theorem for finite W-algebras

Let $\mathfrak{g}$ be the Lie algebra under consideration, let $e\in\mathfrak{g}$ be nilpotent, and let $U(\mathfrak{g},e)$ denote the associated modular finite $W$-algebra. Theorem

$$asserts the existence of the relevant codimension-one ideal in the rigid nilpotent cases. **Codimension-one ideal conjecture.** Theorem$$

should hold for all nilpotent elements $e\in\mathfrak{g}$.

The conjecture extends the paper's codimension-one ideal result from rigid nilpotent elements to arbitrary nilpotent elements, which would extend the resulting conclusions about small modules and one-dimensional representations. The source does not specify whether this extension has been resolved.