The characterization conjecture for positive representations

Let $U_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$ be the modular double of the split real simply-laced quantum group. Consider representations satisfying the positivity and transcendental relations associated with the positive representations. A positive representation is a representation in this class, and tensor products are understood in the direct-integral sense.

Characterization conjecture. The positivity and transcendental relations completely characterize the class of positive representations for $U_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$. In particular, the class of positive representations is closed under tensor product as a direct integral.

For $U_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$, closure under tensor products is known. The corresponding characterization and closure statement for general $\mathfrak{g}$ is posed as a natural question and remains open.