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The abelian and monoidal structure of the category of smooth weight modulesover a non-integrable affine vertex algebra of rank greater than one is aninteresting, difficult and essentially wide open problem. Even conjectures arelacking. This work details and tests such a conjecture for$L_{-\frac{3}{2}}(\mathfrak{sl}_3)$ via a logarithmic Kazhdan-Lusztigcorrespondence.  We first investigate the representation theory of$\overline{U}^H_i(\mathfrak{sl}_3)$, the unrolled restricted quantum group of$\mathfrak{sl}_3$ at fourth root of unity. In particular, we analyse itsfinite-dimensional weight category, determining Loewy diagrams for allprojective indecomposables and decomposing all tensor products of irreducibles.Our motivation is that this category is conjecturally braided tensor equivalentto a category of $W_{A_2}^0(2)$-modules. Here, $W_{A_2}^0(2)$ is an orbifold ofthe octuplet vertex algebra $W_{A_2}(2)$ of Semikhatov, the latter being thenatural $\mathfrak{sl}_3$-analogue of the well known triplet algebra. Moreover,$W_{A_2}^0(2)$ is the parafermionic coset of the affine vertex algebra$L_{-\frac{3}{2}}(\mathfrak{sl}_3)$.  We formulate an explicit conjecture relating the representation theory of$W_{A_2}^0(2)$ and $\overline{U}^H_i(\mathfrak{sl}_3)$ and work out theresulting structures of the corresponding$L_{-\frac{3}{2}}(\mathfrak{sl}_3)$-modules. In particular, we obtainconjectural Loewy diagrams for the latter's projective indecomposables anddecompositions for the fusion products of its irreducibles. These productscoincide with those recently computed via Verlinde's formula. Finally, we giveanalogous results for $W_{A_2}(2)$.

A Kazhdan-Lusztig correspondence for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$