Lusztig braid-group cluster-map conjecture for parabolic positive representations

Let $\mathcal{P}_\lambda^J$ be a parabolic positive representation, let $\mathcal{X}_q^{\mathbf{D}(\overline{\mathbf{i}})}$ be its associated doubled quantum torus, and let $\mathbf{e}_i$ and $\mathbf{f}_i$ denote the corresponding quantum-group generators. Lusztig braid-group cluster-map conjecture. There exists a cluster map of $\mathcal{X}_q^{\mathbf{D}(\overline{\mathbf{i}})}$ for every parabolic positive representation $\mathcal{P}_\lambda^J$ that interchanges the actions of $\mathbf{e}_i$ and $\mathbf{f}_i$. Such an interchange has been verified in a few simple cases, while the general mutation sequence is described as complicated and its factorization in the parabolic case remains unclear. The source also states that the universal $\mathcal{R}$ operator would consequently be well-defined as a unitary transformation on tensor products of parabolic positive representations.