  The abelian and monoidal structure of the category of smooth weight modules
over a non-integrable affine vertex algebra of rank greater than one is an
interesting, difficult and essentially wide open problem. Even conjectures are
lacking. This work details and tests such a conjecture for
$L_{-\frac{3}{2}}(\mathfrak{sl}_3)$ via a logarithmic Kazhdan-Lusztig
correspondence.
  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 its
finite-dimensional weight category, determining Loewy diagrams for all
projective indecomposables and decomposing all tensor products of irreducibles.
Our motivation is that this category is conjecturally braided tensor equivalent
to a category of $W_{A_2}^0(2)$-modules. Here, $W_{A_2}^0(2)$ is an orbifold of
the octuplet vertex algebra $W_{A_2}(2)$ of Semikhatov, the latter being the
natural $\mathfrak{sl}_3$-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 the
resulting structures of the corresponding
$L_{-\frac{3}{2}}(\mathfrak{sl}_3)$-modules. In particular, we obtain
conjectural Loewy diagrams for the latter's projective indecomposables and
decompositions for the fusion products of its irreducibles. These products
coincide with those recently computed via Verlinde's formula. Finally, we give
analogous results for $W_{A_2}(2)$.
