Tensor-product closure of positive representations

Let $\mathcal{P}_\lambda$ and $\mathcal{P}_\mu$ be positive representations of the split real quantum group $\mathcal{U}_{q\widetilde{q}}(\mathfrak{g}_{\mathbb{R}})$, and let $\textbf{C}_k$ be the generalized Casimir operators associated with the fundamental representations $V_k$, for $k=1,\ldots,n$. The coproduct $\Delta(\textbf{C}_k)$ acts on $\mathcal{P}_\lambda\otimes\mathcal{P}_\mu$. Tensor-product closure conjecture. The positive representations $\mathcal{P}_\lambda$ are closed under taking tensor products. Consequently, the coproduct $\Delta(\textbf{C}_k)$ acting on $\mathcal{P}_\lambda\otimes\mathcal{P}_\mu$ is a positive operator with spectrum bounded below by $\dim V_k$ for every $k=1,\ldots,n$, and the coproducts can be simultaneously diagonalized. This conjecture would extend the positivity and spectral properties of generalized Casimir operators from individual positive representations to their tensor products; the source gives no evidence of a resolution.