Categories over quantum affine algebras and monoidal categorification

Let (Formula presented) be a quantum affine algebra of untwisted affine ADE type, and (Formula presented) the Hernandez-Leclerc category of finite-dimensional (Formula presented)-modules. For a suitable infinite sequence (Formula presented) of simple reflections, we introduce subcategories (Formula presented) of (Formula presented) for all (Formula presented). Associated with a certain chain (Formula presented) of intervals in [a, b], we construct a real simple commuting family (Formula presented) in (Formula presented), which consists of Kirillov-Reshetikhin modules. The category (Formula presented) provides a monoidal categorification of the cluster algebra (Formula presented), whose set of initial cluster variables is (Formula presented). In particular, this result gives an affirmative answer to the monoidal categorification conjecture on (Formula presented) by Hernandez-Leclerc since it is (Formula presented), and is also applicable to (Formula presented) since it is (Formula presented).