Uniform boundedness conjecture for super-transitivity of type A étale algebras

Let $N$ and $k$ range over positive integers, and consider all non-pointed étale algebra objects in the categories $\mathcal{C}(\mathfrak{sl}_N,k)$. Their super-transitivity is the invariant appearing in the classification problem. Uniform boundedness conjecture. There exists $K\in\mathbb{N}$ such that every non-pointed étale algebra object in every category $\mathcal{C}(\mathfrak{sl}_N,k)$ has super-transitivity at most $K$. This is a weaker conjecture intended to parallel the corresponding boundedness conjecture for subfactors; the source does not report a resolution.