Relative asymptotic multiplicity conjecture for simple Lie algebras

Let $\overline{\mathfrak{g}}$ be a finite-dimensional simple Lie algebra, and let $\lambda\in \overline{P}_+\backslash\{0\}$ be a non-zero dominant integral weight. The relative asymptotic multiplicity condition is the condition referred to in the source as equation

. **Relative asymptotic multiplicity conjecture.** For all finite-dimensional simple Lie algebras $\overline{\mathfrak{g}}$ and all non-zero dominant integral weights $\lambda\in \overline{P}_+\backslash\{0\}$, the relative asymptotic multiplicity condition

is satisfied. This conjecture would establish the relative asymptotic multiplicity property needed to match the large-colour limits of coloured Jones polynomials with the corresponding $\mathscr{W}$-algebra characters. No resolution is given here.