Partial Kazhdan–Lusztig correspondence for $W^0_{\!A_2}(2)$

Let $\mathscr{E}$ be the category formed by the $W^0_{\!A_2}(2)$-modules obtained by decomposing the $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$-modules in $\mathscr{D}$. Let $\mathscr{C}_{\mathbb{R}}$ be the full subcategory of finite-dimensional weight $\overline{\mathcal{U}}{}_{\mathsf{i}}^H\!(\mathfrak{sl}_3)$-modules with real weights, and let $E_{\lambda}^{(8)}$ and $M_{\varphi(\lambda)+\rho}$ denote the corresponding modules. Partial Kazhdan–Lusztig correspondence. The category $\mathscr{E}$ forms a vertex tensor category, and the identification of $E_{\lambda}^{(8)}$ with $M_{\varphi(\lambda)+\rho}$ partially defines a braided tensor equivalence between $\mathscr{E}$ and $\mathscr{C}_{\mathbb{R}}$. This is explicitly presented as a partial correspondence, while the later statement proposes its extension to all of $\mathscr{C}_{\mathbb{R}}$; no resolution is supplied.