The quantum cluster structure conjecture for quantum unipotent cells

Let $W$ be the Weyl group of a symmetrizable Cartan datum, let $w\in W$, and let $\mathcal{A}_v[N^w]$ denote the corresponding quantum unipotent cell. Quantum cluster structure conjecture. For any Weyl group element $w\in W$ there is a quantum cluster structure on $\mathcal{A}_v[N^w]$, where cluster monomials correspond to irreducible characters of KLR representations. This proposes a cluster-algebraic realization of irreducible KLR characters for quantum unipotent cells, extending the established symmetric-type relationship with the dual canonical basis.