Existence and uniqueness of double quantum affinizations for untwisted affine Kac–Moody algebras

Let $\dot{\mathfrak g}$ be an untwisted affine Kac–Moody Lie algebra, and let $\ddot{\mathrm{U}}_q(\dot{\mathfrak g})$ denote its double quantum affinization. Existence and uniqueness conjecture. Every untwisted affine Kac–Moody Lie algebra $\dot{\mathfrak g}$ admits a unique up to isomorphism double quantum affinization $\ddot{\mathrm{U}}_q(\dot{\mathfrak g})$. This is presented as a further natural extension of the type ${\mathfrak a}_1$ construction, and no general proof or resolution is given in the source.