Flag-tuple parametrization 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 $\OqG$ be the standard quantum group with shape algebra. Let $T^\mathrm{DJ}=(E^\mathrm{DJ},\sig_1,\dots,\sig_\ell,\L_1,\dots,\L_\ell)$ be the tuple constructed in the paper, and let a flag module mean a module of the shape algebra arising from the corresponding geometric construction.

Flag-tuple parametrization conjecture. The tuple $T^\mathrm{DJ}$ is the flag tuple associated to $\OqG$; equivalently, $E^\mathrm{DJ}$ parametrizes all flag modules of the shape algebra of $\OqG$.

Together with the preceding reconstruction conjecture, this asserts that the proposed geometric tuple captures not only the shape algebra but also all of its flag modules. The supplied text does not state a proof or resolution of this assertion, so its status remains open.