The quantum-group si]-system scalar-splitting conjecture

Let $\mathcal{U}_\varpi=U_\varpi(\mathfrak{g})$ be a quantized universal enveloping algebra at an odd root of unity, and let $\mathcal{B}^{\pm}_\varpi$ be its Borel Hopf subalgebras. The associated Cayley--Hamilton Hopf algebras produce a monoidal Ab-category $\mathcal{C}$ with a \Psi-system. Let $H=\widehat H\oplus\check H$, and let $q^{}\in\operatorname{End}(H)$ be the operator from Lemma 17. Quantum-group \widehat{\Psi}-system scalar-splitting conjecture. This \Psi-system can be extended to a \widehat{\Psi}-system in $\mathcal{C}$ such that there is a scalar $\widetilde q\in\mathbb{C}$ for which $q^{}$ is $T$-equal to

$$\widetilde q\operatorname{Id}_{\widehat H}\oplus\widetilde q^{-1}\operatorname{Id}_{\check H}.$$

This is the specific scalar condition needed in the construction of the 3-manifold invariants; the supplied text does not state whether this assertion has been proved beyond the surrounding construction.