Categorical Grothendieck-ring conjecture for the shifted quantum affine cluster algebra

Let $\mathcal{A}$ be the infinite-rank cluster algebra and let

$$F:\mathcal{A}\longrightarrow K_0(\mathcal{O}^{\mathrm{sh}}_{\mathbb{Z}})$$

be the injective ring morphism described above. Let $\mathcal{O}^{\mathrm{sh,f}}_{\mathbb{Z}}$ be the full subcategory of finite-length representations in $\mathcal{O}^{\mathrm{sh}}_{\mathbb{Z}}$, so that $K_0(\mathcal{O}^{\mathrm{sh,f}}_{\mathbb{Z}})$ is a subring of the topological Grothendieck ring. Categorical Grothendieck-ring conjecture. The ring morphism $F$ induces a ring isomorphism

$$\mathcal{A}\simeq K_0(\mathcal{O}^{\mathrm{sh,f}}_{\mathbb{Z}}).$$

This gives a finer categorical interpretation of the cluster algebra by identifying it with the Grothendieck ring of finite-length objects in the shifted representation category. It relies on the preceding expected simplicity of the images of cluster variables and remains open.