Fused-current commutation conjecture for minimal pairs

Let $\Delta^+$ be the positive roots of a finite-type ADE Lie algebra with a fixed total convex order, and let $\widetilde{f}_\alpha(x)=\sum_{d\in\mathbb{Z}}\widetilde{f}_{\alpha,d}/x^d$ be the fused currents. For a minimal pair $\alpha<\beta$, let $u,v\in\mathbb{Z}$ and $c(q)\in\mathbb{Z}[q,q^{-1}]^\times$ depend on $\alpha$ and $\beta$, and write $\delta(x)=\sum_{d\in\mathbb{Z}}x^d$. Fused-current commutation conjecture. For every minimal pair $\alpha<\beta$,

$$\widetilde{f}_{\alpha}(x)\widetilde{f}_{\beta}(y)-\widetilde{f}_{\beta}(y)\widetilde{f}_{\alpha}(x)\frac{xq^v-yq^{(\alpha,\beta)}}{xq^{v+(\alpha,\beta)}-y}=c(q)\,\delta\left(\frac{xq^{v+(\alpha,\beta)}}{y}\right)\widetilde{f}_{\alpha+\beta}(xq^u).$$

In the second term on the left, the rational function is expanded in the region $|x|\gg|y|$. This is a proposed analogue of the Levendorskii–Soibelman commutation relations for fused currents; its specialization at $q=1$ is intended to recover the loop-algebra bracket.