The classification conjecture for 1-super-transitive étale algebras in type A

Let $N$ and $k$ range over positive integers, and let $\mathcal{C}(\mathfrak{sl}_N,k)$ denote the corresponding categories. Let $\mathcal{A}$ be the set of all non-pointed étale algebra objects in these categories, and let super-transitivity be the invariant used to measure such objects. Classification conjecture. There exist only finitely many $A\in\mathcal{A}$ with super-transitivity strictly greater than $1$. Moreover, if $A\in\mathcal{A}$ is 1-super-transitive, then $A$ is one of the étale algebras coming from one of the conformal embeddings

$$\mathcal{V}(\mathfrak{sl}_{N},N-2)\subseteq\mathcal{V}(\mathfrak{sl}_{N(N-1)/2},1),$$ $$\mathcal{V}(\mathfrak{sl}_{N},N)\subseteq\mathcal{V}(\mathfrak{so}_{N^2-1},1),$$ $$\mathcal{V}(\mathfrak{sl}_{N},N+2)\subseteq\mathcal{V}(\mathfrak{sl}_{N(N+1)/2},1),$$

or a simple current extension of one of these embeddings. This would classify all étale algebras in the categories $\mathcal{C}(\mathfrak{sl}_N,k)$ if proved constructively; the conjecture is presented as a refinement of an earlier conjecture by the first author and Noah Snyder.