Inclusion conjecture for quantum Grothendieck rings of categories O and C

Let $\mathscr{C}_{\mathbb{Z}}$ be the finite-dimensional representation category and let $\mathcal{O}^+_{\mathbb{Z}}$ be the category whose quantum Grothendieck ring is $K_t(\mathcal{O}^+_{\mathbb{Z}})$. Let $K_t(\mathscr{C}_{\mathbb{Z}})$ be the existing quantum Grothendieck ring, and let

$$\mathcal{J}:\mathscr{Y}_t\hookrightarrow\mathscr{T}_t$$

be the injective morphism extending to the ambient quantum tori. Inclusion conjecture. The morphism $\mathcal{J}$ restricts to an inclusion

$$\mathcal{J}:K_t(\mathscr{C}_{\mathbb{Z}})\subset K_t(\mathcal{O}^+_{\mathbb{Z}}).$$

This would identify the quantum Grothendieck ring of the finite-dimensional subcategory with a subring of the larger quantum Grothendieck ring; the source presents it as a natural desired property and does not establish it in general.