Cluster monomial conjecture for real simple modules

Let $\mathcal{A}$ be the cluster algebra identified with the ring of truncated $q$-characters of the category $\mathcal{C}^-$. A simple $U_q(\widehat{\mathfrak{g}})$-module $S$ is real if $S\otimes S$ is simple. Cluster monomial conjecture. Under the identification of $\mathcal{A}$ with the ring of truncated $q$-characters of $\mathcal{C}^-$, the cluster monomials correspond exactly to the truncated $q$-characters of the real simple modules of $\mathcal{C}^-$. This connects cluster-monomial combinatorics with real simple modules; in types $A$, $D$, and $E$, the source says it is essentially equivalent to a conjecture of Hernandez and Leclerc, while the status in the stated generality is not resolved here.