Categories over quantum affine algebras and monoidal categorification

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