Branching rules for coset construction: $widehat{su}(4)_2 rightarrow frac{widehat{su}(4)_2}{widehat{su}(2)_4} oplus widehat{su}(2)_4$

I am trying to find the branching rules for the coset construction: $widehat{su}(4)_2 rightarrow frac{widehat{su}(4)_2}{widehat{su}(2)_4} oplus widehat{su}(2)_4$, where the subscript indicates the level of the affine Lie algebra, and the coset model is $frac{widehat{su}(4)_2}{widehat{su}(2)_4}$.
If it is helpful, the central charges are: $c_{widehat{su}(4)_2} = 5$, $c_{widehat{su}(2)_4} = 2$, and $c_{frac{widehat{su}(4)_2}{widehat{su}(2)_4}} = 3$.

If anyone knows the branching rules, or knows a procedure by which this can be computed, it would be much appreciated.
(As an example of something I am looking for, is Table I in https://arxiv.org/abs/hep-th/9309093, where the branching rules for the maverick coset $frac{widehat{su}(3)_2}{widehat{su}(2)_8}$ are listed.)