A finite-dimensional module $L$ over an affine quantum group is called real if the tensor product $L \otimes L$ is irreducible. It is known that for an appropriately chosen subcategory $\mathcal{C}$ of the category of finite-dimensional representations, the　Grothendieck ring $K(\mathcal{C})$ has a cluster algebra structure in which cluster monomials correspond to irreducible modules (categorification of cluster algebras). In such a category $\mathcal{C}$, Hernandez-Leclerc conjectured that an irreducible module corresponds to a cluster monomial if and only if it is a real module. This conjecture remains generally open, except for cases where the categorified cluster algebra is of finite type. In this talk, I will construct subcategories that categorify cluster algebras of affine type and explain that the conjecture holds within these categories.