Hernandez–Leclerc's bijection conjecture for cluster monomials and real simple modules

Let $\mathscr{C}_{\ell}$ be the Hernandez–Leclerc monoidal subcategory and let $K_{0}(\mathscr{C}_{\ell})$ be its Grothendieck ring, identified with a cluster algebra. A simple module is real if its tensor square is simple, and a real simple module is real prime if it cannot be written as a nontrivial tensor product of simple modules. Hernandez–Leclerc's bijection conjecture. There is a bijection between the cluster monomials (respectively, cluster variables) in $K_{0}(\mathscr{C}_{\ell})$ and real simple (respectively, real prime simple) modules in $\mathscr{C}_{\ell}$. This conjecture predicts a precise correspondence between cluster-algebraic objects and simple modules; the supplied text gives no evidence that it has been resolved.