Asymptotic dimensionality for identical angular-momentum values

Let ${\bm l}=(l,\ldots,l)\in\mathcal{L}^N$ contain identical nonzero values $l$, and let $L\in\mathcal{L}_G$ satisfy $L+\sum {\bm l}\in\mathbb{Z}$. Define

$$\operatorname{Var}_{\bm l}=\frac{Nl(N+2l+1)}{6}.$$

Asymptotic dimensionality conjecture. If $L\ll\sqrt{\operatorname{Var}_{\bm l}}$, then

$${{\rm \dim}}({\bar V}^{{\bm l},L})=\binom{N+2l}{2l}\left(\frac{2L+1}{2\sqrt{2\pi}\,\operatorname{Var}_{\bm l}^{3/2}}+O\left(\frac{1}{N^{5/2}}\right)\right).$$

The quantity ${{\rm \dim}}({\bar V}^{{\bm l},L})$ counts the relevant geometric-equivariant, permutation-invariant states. The estimate is motivated by numerical results for identical ${\bm l}$ and by asymptotic results on counting coefficients of Gaussian binomial coefficients, but the authors state that they were unable to prove it.