Berenstein–Zelevinsky's quantum twist automorphism conjecture

Let $\mathbf{i}\in I^m$ be a reduced word for $w\in W$, let $\mathcal L_\mathbf{i}$ be the associated quantum torus, let $\mathcal U_\mathbf{i}$ be its quantum cluster algebra, and let $\hat\eta_\mathbf{i}:\mathcal L_\mathbf{i}\to\operatorname{Frac}(\mathcal L_\mathbf{i})$ be defined by $X_k\mapsto\hat X_k$, where

$$\hat X_k=\Psi_\mathbf{i}(\Delta_{s_{i_1}\cdots s_{i_k}\omega_{i_k}}).$$

Berenstein–Zelevinsky's twist conjecture. Under the hypotheses of the commutation lemma, the restriction of $\hat\eta_\mathbf{i}$ to $\mathcal U_\mathbf{i}$ is an isomorphism of algebras

$$\eta_\mathbf{i}:\mathcal U_\mathbf{i}\to\mathcal U_\mathbf{i}.$$

This is identified as a particular case of Berenstein–Zelevinsky's Conjecture 10.10; the supplied text does not claim a proof in general.