Nakajima's monoidal categorification conjecture for ADE cluster algebras

Let $\mathscr A$ be the cluster algebra for the quiver defined in Section, and suppose that its underlying graph is of type $ADE$. Let $\mathscr C_1$ denote the explicitly defined monoidal subcategory of finite-dimensional representations of the quantum affine algebra $\mathbf U_q(\mathbf L\frak g)$.

Nakajima's monoidal categorification conjecture. The cluster algebra $\mathscr A$ has a monoidal categorification, given by the monoidal subcategory $\mathscr C_1$.

This is a proposed concrete realization of Hernandez–Leclerc's categorical approach in simply laced finite type. The source does not provide a resolution status for this assertion.