Geiss–Hernandez–Leclerc's cluster categorification conjecture for shifted category O

Let $\mathcal{O}_{sh}^{\mathbb Z}$ be the integral finite-length subcategory of shifted category $\mathcal{O}_{sh}$, and let $\mathscr A$ be the cluster algebra associated with the extended $QQ$-system. Geiss–Hernandez–Leclerc's conjecture. The isomorphism between the completed ambient algebra and the completed cluster algebra restricts to an isomorphism

$$K_0(\mathcal{O}_{sh}^{\mathbb Z})\cong \mathscr A.$$

Equivalently, the integral shifted category $\mathcal{O}_{sh}^{\mathbb Z}$ should provide a monoidal categorification of $\mathscr A$. This is the paper's main motivating conjecture; the authors make progress towards it, but the full isomorphism remains open in the supplied text.