Semisimplicity conjecture for the category $KL_k$

Let $\mathfrak{g}$ be the affine Lie superalgebra considered in the paper, let $k$ be its level, and let $KL_k$ denote the corresponding category of modules. The category $KL_k$ is semisimple when every object is a direct sum of simple objects.

Semisimplicity conjecture. The category $KL_k$ is semisimple if and only if

$$k \in \left\{-1,-\frac{m+1}{m+2}\ \middle|\ m\in\mathbb{Z}_{\geq 0}\right\}.$$

This is expected based on the paper's semisimplicity theorem and its analysis of admissible levels. The supplied text does not state that the conjecture has been proved or disproved.