Hernandez–Leclerc's cluster-monomial conjecture for real simple modules

Let $\mathscr{C}^-$ be the subcategory of finite-dimensional $U_q(\widehat{\mathfrak{g}})$-modules described above, and let a simple object be real when its tensor square is simple. The cluster algebra structure on the Grothendieck ring of $\mathscr{C}^-$ has cluster monomials and real simple objects as its relevant objects.

Hernandez–Leclerc's conjecture. The cluster monomials of the cluster algebra are in bijection with the isomorphism classes of real simple objects in $\mathscr{C}^-$.

This conjecture concerns the proposed monoidal categorification of the cluster algebra associated with $\mathscr{C}^-$. Earlier results establish analogous categorification and correspondence statements for important subcategories and Dynkin types, but the source provides no resolution of this conjecture.