Non-full exactness conjecture for vector spaces over general small quantum groups

Let $\mathcal{C}$ be the braided tensor category of finite-dimensional modules over $u_q(\mathfrak{g})$, where $q$ is a root of unity of odd order and the category is equipped with the standard factorizable R-matrix. Let $\mathbf{vect}$ be the canonical $\mathcal{C}$-module category. Non-full exactness conjecture. The $\mathcal{C}$-module category $\mathbf{vect}$ is not fully exact. This generalizes the proposition established for $u_q(\mathfrak{sl}_2)$; the claim for general $\mathfrak{g}$ remains open.