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

Let $\mathscr{E}$ be the vertex tensor category of $W^0_{\!A_2}(2)$-modules obtained by decomposing the relevant $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$-modules, and 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. For generic $\lambda$ and the specified parameter regions, use the modules $E_{\lambda}^{(d)}$, $L_{\varphi(\lambda)+\rho}^{(d)}$, and $L_{\varphi(\lambda)}^{(d)}$ appearing below. Full Kazhdan–Lusztig correspondence. There is a braided tensor equivalence between $\mathscr{E}$ and $\mathscr{C}_{\mathbb{R}}$ defined by the displayed assignments for $d=8,4,3,\overline{3},1$, with the exact genericity and parameter conditions stated in the source. The proposed equivalence extends the partial correspondence to atypical modules of degrees $1$, $3$, $4$, and $8$; the supplied text does not give evidence that it has been proved or refuted.