4 problems

Strong cluster-map conjecture. For every $w\in W$, the coordinate ring $\mathbb{C}[U^w]$ is a cluster algebra modelled by the strong cluster map $\varphi_w$ for the standard compon…

The cluster structure conjecture. The coordinate ring $\mathbb{C}[U^w]$ should have a cluster algebra structure modelled by the corresponding stably 2-Calabi–Yau category.

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 al…

Let $A$ be a symmetrizable $I\times I$ Cartan matrix, let $\mathbf{i}=(i_1,\ldots,i_m)$ be a reduced word for $w\in W$, and let $\mathcal{U}_\mathbf{i}$, $\mathcal{L}_\mathbf{i}$,…