Geometric reconstruction conjecture for standard Drinfeld–Jimbo quantum groups

Let $G$ be a simply-connected simple complex group, let $q\in\C^*$ be not a root of unity, and let $T^\mathrm{DJ}=(E^\mathrm{DJ},\sig_1,\dots,\sig_\ell,\L_1,\dots,\L_\ell)$ be the tuple defined from the geometric data in the paper. Let $M(T^\mathrm{DJ})$ be its associated quadratic algebra, and let $\OqG$ denote the standard quantum group.

Geometric reconstruction conjecture. The tuple $T^\mathrm{DJ}$ is a braided tuple, and

$$M(T^\mathrm{DJ})$$

is the shape algebra of $\OqG$.

This conjecture gives a positive answer to the question of whether a quadratic shape algebra of a standard quantum group can be reconstructed from a braided tuple. The source states that the assertion is proved for $G=\SL(n)$ later in the paper, while it is conjectural for general simply-connected simple complex groups.