Character-transfer conjecture from twisted to untwisted Borel modules

Let w\frac{in W, let $bpsi=T_w(bpsi_{i,a}^{-1})$, and write the projected limit $bpsipi_{q,\infty}^w$ as $c\times a$, with $c$ its constant part and $a$ its non-constant part. Let $w_0$ be the longest element of $W$, and let $c^{-1}$ denote the image of $c$ under $e^\alpha\to e^{-\alpha}$. Character-transfer conjecture. The product $c^{-1}\times a$ belongs to the untwisted completed character ring and equals

$$c^{-1}\times a=\chi_q(L(T_w(bpsi_{i,a}^{-1}))).$$

This conjecture proposes a relation between the $q$-characters of modules in $\mathcal{O}^w$ and $\mathcal{O}$; the source says it has been verified in some examples.