Ding–Khoroshkin completion conjecture for fused currents

Let $U_q(L\mathfrak{g})$ be the quantum loop group of a finite-type ADE Lie algebra, with negative half $U_q(L\mathfrak{n}^-)$ and a completion $\widehat{U}_q(L\mathfrak{n}^-)$, and let $\Delta^+$ be the set of positive roots. For each $\alpha\in\Delta^+$ and $d\in\mathbb{Z}$, let $\widetilde{f}_{\alpha,d}$ be the coefficient of the fused current $\widetilde{f}_\alpha(x)=\sum_{d\in\mathbb{Z}}\widetilde{f}_{\alpha,d}/x^d$, and let $\widehat{U}_q(L\mathfrak{n}^-)_{-\alpha,d}\to\widehat{U}_q(\widehat{\mathfrak{g}})$ be the algebra homomorphism induced by the compatible completions. Ding–Khoroshkin completion conjecture. For every $(\alpha,d)\in\Delta^+\times\mathbb{Z}$,

$$\widetilde{f}_{\alpha,d}\in\operatorname{Im}\Big(\widehat{U}_q(L\mathfrak{n}^-)_{-\alpha,d}\longrightarrow\widehat{U}_q(\widehat{\mathfrak{g}})\Big).$$

This would give a formal algebraic realization of the fused currents, whose original construction used countable sums defined only in suitable representations. The paper states that this claim is expected to be proved in future work.