Affine Weyl group description conjecture for the map to the fusion ring

Let $P$ be the weight lattice, $P^+$ the dominant integral weights, $P_{\ell}\subset P^+$ the weights of level $ell$, and let $W_{ell}$ be the affine Weyl group acting on the weight lattice. Let $\rho$ be the half sum of the positive roots, let $ve:W_{ell}to\{pm1\}$ be the signature homomorphism, and let $pi_{\kappa}:\mathcal{FR}_{\kappa}(\mathcal{Fg})to\mathcal{FF}_{\kappa}(\mathcal{Fg})$ be the previously defined algebra homomorphism. Affine Weyl group map conjecture. For every dominant integral weight $\lambda$, $pi_{\kappa}([\lambda])=0$ if and only if no $win W_{ell}$ satisfies $w(\lambda+\rho)-rhoin P_{ell}$. If $w(\lambda+\rho)-\rho=muin P_{ell}$ for some $w$, then

$$\pi_{\kappa}([\lambda])=\ve(w)p_{\kappa}(\lambda)[\mu],$$

where $p_{\kappa}(\lambda)$ is a monomial in $t$. This conjecture seeks an intrinsic description of the map to the fusion ring; the paper then proposes more precise formulas for the monomial in special cases, but gives no proof of the general assertion.