Frenkel–Hernandez character expansion conjecture

Assume that ${\mathfrak{g}}$ is of finite type. Let $W$ be the Weyl group, let $i\in I$, let $a\in\mathbb{C}^*$, and let $\boldsymbol{\psi}_{w(\omega_i^\vee),a}$ be the corresponding $\ell$-weight. Write $\chi_{w(\omega_i^\vee)}$ for the series introduced in the cited work, and let $\chi^{w(\omega_i^\vee)}$ be the normalization factor. Frenkel–Hernandez character expansion conjecture. For every $w\in W$, $i\in I$, and $a\in\mathbb{C}^*$,

$$\chi(L(\boldsymbol{\psi}_{w(\omega_i^\vee),a}))=\chi^{w(\omega_i^\vee)}\chi_{w(\omega_i^\vee)}.$$

The source states that this is established for the simple reflection case $w=s_i$, but gives no resolution for general $w$.