The ribbon-category conjecture for the full category of non-restricted quantum-group modules

Let $\mathcal{D}^{H}$ be the category of modules under consideration, and let $\varpi\in\mathcal{U}^{H}$ be the central element defined by $\varpi=uS(u)^{-1}K_{2\rho}^{2(r-1)}$. Let $\mathcal{D}^{\theta}$ be the full subcategory of $\mathcal{D}^{H}$ consisting of modules on which $\varpi$ acts as the identity. Ribbon-category conjecture. One has $\varpi=1$ and

$$\mathcal{D}^{H}=\mathcal{D}^{\theta}$$

and this category is a ribbon category. The preceding theorem establishes a ribbon structure on the subcategory $\mathcal{D}^{\theta}$; the conjecture asserts that the restriction imposed by $\varpi$ is unnecessary for the full category.