The monoidal reachability conjecture

Let $\mathscr{C}_\ell$ be the Hernandez–Leclerc category for $\ell\in\mathbb{Z}_{\geq 1}$, let $\mathcal{P}_\ell^+$ be its monoid of dominant monomials, and let $K_0(\mathscr{C}_\ell)$ carry the cluster algebra structure described in the source. For a dominant monomial $M$, write $L(M)$ for the associated simple module and $\chi_q(L(M))$ for its $q$-character. Monoidal reachability conjecture. For every $\ell\in\mathbb{Z}_{\geq 1}$, the $q$-character of every real prime module, respectively every real module, in $\mathscr{C}_\ell$ is a cluster variable, respectively a cluster monomial, in $K_0(\mathscr{C}_\ell)$. The converse direction of the Hernandez–Leclerc conjecture is explicitly described as widely open in the source.