GHL monoidal categorification conjecture for shifted category O

Assume that ${\mathfrak{g}}$ is simply laced. Let $\mathcal{A}$ be the infinite-rank cluster algebra embedded in $K_0(\mathcal{O}^{\mathrm{sh}})$ by

$$i:\mathcal{A}\longrightarrow K_0(\mathcal{O}^{\mathrm{sh}}),$$

and let $I^{-1}\circ i$ be the induced embedding into $K_0(\mathcal{O})$. Let $\overline{K}_0(\mathcal{O})$ denote the renormalized Grothendieck ring, and let $\chi^\mu$ be the normalization factor associated with a shift $\mu$. GHL monoidal categorification conjecture. The image by $I^{-1}\circ i$ of every cluster monomial in $\mathcal{A}$ should be a simple class up to a factor $\chi^\mu$, and should belong to $\overline{K}_0(\mathcal{O})$. This is a reformulation of the monoidal categorification conjecture for shifted category $\mathcal{O}$; the source presents it as conjectural, and no resolution is supplied.