Explicit coefficient 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 with length function $l$. Let $\rho$ be the half sum of the positive roots, and write a weight $xi$ as $\sum_i a_ialpha_i$ in terms of simple roots, defining $s(xi)=\sum_i a_i$. Suppose $\lambda$ is dominant integral and $w(\lambda+\rho)-\rho=muin P_{ell}$ for some $win W_{ell}$. Explicit coefficient conjecture. If $\kappa=ell+h^{vee}$, then

$$\pi_{\kappa}([\lambda])=\ve(w)t^{s(\lambda-\mu)}[\mu].$$

Moreover, in the conjugate setting,

$$\pi_{\bar{\kappa}}([\lambda])=\ve(w)t^{l(w)}[\mu]$$

when $l(w)$ is minimal among all such $w$. These formulas refine the general affine Weyl group description of the map to the fusion ring and are presented as conjectures for the two specified cases; no proof or resolution is supplied.