Conjecture on the quantization matrix for double Bruhat cells

Let $G$ be a simply-connected, simple, complex algebraic group with Lie algebra $\mathfrak{g}$ of simply-laced type. Let $\mathcal{A}_{t,w_0}$ be the quantum cluster algebra structure on the coordinate ring of the double Bruhat cell $G^{w_0,w_0}$, and let $\Lambda_0$ and $\bar{\Lambda}_c$ be the quantization matrices defined from the preceding examples. Quantization-matrix conjecture. The quantum cluster algebra structure induced by $\mathcal{A}_{t,w_0}$ is, up to rescaling, the same as the structures of Berenstein–Zelevinsky and Yakimov on the corresponding quantum coordinate ring. More precisely, there exists an integer $k$ such that

$$\Lambda_0=k\bar{\Lambda}_c.$$

The conjecture proposes compatibility between these quantum cluster structures for every simply-laced $G$; its resolution is not specified in the source.