Constancy of the leading-coefficient ratio for Gelfand–Kirillov subquotients

Let $\mathfrak{g}$ be the Lie algebra under consideration, let $d$ be fixed, and let $\mathcal{O}_\lambda^d$ be the category of modules specified in the paper. For $M\in\mathcal{O}_\lambda^d$, let $\operatorname{LC}(M)$ denote the leading coefficient of the quasi-polynomial from Proposition 4.1, and let $\underline{\operatorname{LC}}(M)$ denote the corresponding lower leading coefficient.

Leading-coefficient ratio conjecture. There exists a constant $C$, depending only on the Lie algebra $\mathfrak{g}$ and $d$, such that for all $M\in\mathcal{O}_\lambda^d$,

$$\underline{\operatorname{LC}}(M)=C\cdot\operatorname{LC}(M).$$

This conjecture asserts that the two leading-coefficient invariants differ by a universal multiplicative factor within each fixed Lie algebra and degree. The supplied text gives no evidence that the conjecture has been resolved.