Borel-category realization of the generalized QQ-system

Assume that ${\mathfrak{g}}$ is of finite type. Let $I$ index the simple roots, let $W$ be the Weyl group, and let $\mathcal{Q}_{w(\omega_i^\vee),a}$ and $\boldsymbol{\psi}_{w(\omega_i^\vee),a}$ be as above. Let $\chi^{w(\omega_i^\vee)}$ be the normalization factor and let $\overline{K}_0(\mathcal{O})$ be the renormalized Grothendieck ring. Borel-category $QQ$-system conjecture. For every $i\in I$, $a\in\mathbb{C}^*$, and $w\in W$, one should have

$$\mathcal{Q}_{w(\omega_i^\vee),a}=(\chi^{w(\omega_i^\vee)})^{-1}[L(\boldsymbol{\psi}_{w(\omega_i^\vee),a})]$$

in $\overline{K}_0(\mathcal{O})$. This is equivalent, via the Grothendieck-ring isomorphism in the source, to the shifted-category formulation and is described as a more precise conjectural realization in the quantum affine Borel category; no resolution is supplied.