Higher-rank tensor product decomposition conjecture for positive representations

Let $\mathcal{U}_{q\widetilde{q}}(\mathfrak{g}_\mathbb{R})$ be the modular-double quantum group and let $\mathcal{P}_{\alpha}$ and $\mathcal{P}_{\beta}$ be positive representations. Write

$$\overrightarrow{\gamma}=\sum_{\alpha\in\Delta_+}\gamma_\alpha\omega_\alpha,$$

where the sum ranges over all positive roots, $\gamma_\alpha\in\mathbb{R}_+$, and $\omega_\alpha$ are the fundamental weights, with $\omega_\alpha:=\omega_{\alpha_1}+\omega_{\alpha_2}$ when $\alpha:=\alpha_1+\alpha_2$ is not simple. Tensor product decomposition conjecture. The positive representations are closed under tensor products, with decomposition

$$\mathcal{P}_{\alpha}\otimes\mathcal{P}_{\beta}\simeq\int_{\mathbb{R}_+^N}^{\oplus}\mathcal{P}_{\overrightarrow{\gamma}}\,d\mu(\overrightarrow{\gamma}),$$

where

$$d\mu(\overrightarrow{\gamma})=\prod_{\alpha\in\Delta_+}\sinh(2\pi b\gamma_\alpha)\sinh(2\pi b^{-1}\gamma_\alpha)\,d\gamma_\alpha.$$

This conjecture would extend the known rank-one tensor product decomposition to higher rank and provide the continuous analogue of canonical-basis decompositions. The required functional-analytic treatment and the general higher-rank intertwining transformations remain to be established.