Summary: Let \(U_q'(\mathfrak{g})\) be a quantum affine algebra of untwisted affine ADE type, and \(\mathcal{C}_{\mathfrak{g}}^0\) the Hernandez-Leclerc category of finite-dimensional \(U_q'(\mathfrak{g})\)-modules. For a suitable infinite sequence \(\widehat{w}_0= \cdots s_{i_{-1}}s_{i_0}s_{i_1} \cdots\) of simple reflections, we introduce subcategories \(\mathcal{C}_{\mathfrak{g}}^{[a,b]}\) of \(\mathcal{C}_{\mathfrak{g}}^0\) for all \(a \le b \in \mathbb{Z} \sqcup\{\pm \infty \} \). Associated with a certain chain \(\mathfrak{C}\) of intervals in \([a,b]\), we construct a real simple commuting family \(M(\mathfrak{C})\) in \(\mathcal{C}_{\mathfrak{g}}^{[a,b]} \), which consists of Kirillov-Reshetikhin modules. The category \(\mathcal{C}_{\mathfrak{g}}^{[a,b]}\) provides a monoidal categorification of the cluster algebra \(K(\mathcal{C}_{\mathfrak{g}}^{[a,b]})\), whose set of initial cluster variables is \([M(\mathfrak{C})]\). In particular, this result gives an affirmative answer to the monoidal categorification conjecture on \(\mathcal{C}_{\mathfrak{g}}^-\) by Hernandez-Leclerc since it is \(\mathcal{C}_{\mathfrak{g}}^{[-\infty,0]} \), and is also applicable to \(\mathcal{C}_{\mathfrak{g}}^0\) since it is \(\mathcal{C}_{\mathfrak{g}}^{[-\infty,\infty]} \).