In this paper, we propose a conjectural formula for the highest -weight monomial of an arbitrary real module over a simply-laced quantum affine algebra. We verify the conjecture under a multiplicative reachability condition, answering the Hernandez--Leclerc classification problem in monoidal categorifications of cluster algebras under this condition. Moreover, we introduce the notion of cluster modules, generalizing Kirillov--Reshetikhin modules and Hernandez--Leclerc modules as special cases. We prove that cluster modules are reachable real modules, and obtain a system of equations governing q-characters of the prime cluster modules, providing a natural generalization of both the classical T-system relations for Kirillov--Reshetikhin modules and the exchange relations for Hernandez--Leclerc modules.
Bing et al. (Thu,) studied this question.