Quasi-lisse, unique-module, and tensor-category conjecture for $L_{k_m}(D_4)$

Let $L_{k_m}(D_4)$ be the affine vertex algebra of type $D_4$ at level

$$k_m=-6+\frac{4}{2m+1},\qquad m\geq 1.$$

Let $KL_{k_m}(D_4)$ denote the Kazhdan–Lusztig category of modules for $L_{k_m}(D_4)$, and call a module ordinary when it is an irreducible module in the ordinary category.

Conjecture on $L_{k_m}(D_4)$. For each $m\geq1$, $L_{k_m}(D_4)$ is a quasi-lisse vertex algebra, it has a unique irreducible ordinary module, and $KL_{k_m}(D_4)$ is a semisimple braided tensor category.

The first two assertions repeat the conjecture stated in the abstract, while the categorical assertion gives a stronger expected description of the representation theory. The source supplies no evidence that these claims have been resolved.