The law of large numbers conjecture for Demazure modules of affine $b[0msl_2$

Fix a dominant integral weight $\Lambda$ and a sequence $(w^{(k)})$ in $W^{\mathrm{aff}}$ such that $l(w^{(k)}) \to \infty$. Let $\mu^{(k)} \in \operatorname{Meas}(\mathbf{N} \times \mathbf{Z})$ be the joint distribution of the degree and the finite weight in $V_{w^{(k)}}(\Lambda)$. Let $\tilde\mu^{(k)} \in \operatorname{Meas}(\mathbf{R}^2)$ be the distribution obtained from $\mu^{(k)}$ by normalizing to a probability distribution and rescaling the two coordinates individually so that $\operatorname{supp}(\tilde\mu^{(k)})$ just fits into the rectangle $[0,1] \times [-1,1]$. Law of large numbers conjecture. Then, as $k \to \infty$,

$$\tilde\mu^{(k)} \stackrel{\mathrm{w}}{\longrightarrow} \delta_{\left( \frac{\langle c, \Lambda \rangle +2}{3(\langle c, \Lambda \rangle+1)}, 0 \right)},$$

where $c = \alpha_0^\vee + \alpha_1^\vee$ denotes the canonical central element. This conjecture extends the established result for sequences of level-one weights and predicts concentration of the rescaled joint degree–finite-weight distribution at a point whose first coordinate depends only on the level of $\Lambda$, while the finite-weight coordinate converges to zero.