Admissible-level diagonal coset rationality conjecture

Let $\mathfrak g$ be a simple finite-dimensional Lie algebra, let $r$ be a positive integer, and let $k$ be an admissible level for $\widehat{\mathfrak g}$. Define

$$\mathcal C=\operatorname{Com}(L_{k+r}(\mathfrak g),L_k(\mathfrak g)\otimes L_r(\mathfrak g)).$$

Diagonal coset rationality conjecture. The coset $\mathcal C$ is lisse and rational. This generalizes the known simply-laced diagonal-coset cases and the lattice-subalgebra situation, while the assertion for arbitrary simple $\mathfrak g$, positive $r$, and admissible $k$ remains open.