The cluster structure conjecture for Grothendieck rings of quantum affine algebras

Let $\mathcal{C}_{\epsilon^\mathbb{Z}}$ be the monoidal category associated to $\mathfrak{g}=\mathfrak{sl}_k$ at a parameter $\epsilon\in\mathbb{C}^*$ such that $\epsilon^2$ is an $\ell$th root of unity, and let $K_0(\mathcal{C}_{\epsilon^\mathbb{Z}})$ be its Grothendieck group. A generalized upper cluster algebra structure is a cluster-algebra structure allowing generalized exchange relations; its cluster monomials are products of cluster variables taken from a single cluster. The cluster structure conjecture. The Grothendieck group $K_0(\mathcal{C}_{\epsilon^\mathbb{Z}})$ admits a generalized upper cluster algebra structure in which every cluster monomial is the class of a simple module. An initial seed can be obtained by setting all frozen variables $\overline{x}_{N+i}=1$ in Definition~ and identifying the mutable variables $\overline{x}_i$ and coefficient-string variables $z_s$ with the classes of appropriate simple modules. This conjecture extends the known cases for $k=2$ and for $k=3$, $\ell=2$, and the conjectural initial seed previously given for all $k=3$ cases; the general cases remain open.