The classification conjecture for good braid-group representations

Let $B_k$ be the braid group on $k$ strands and let $S_n$ be the symmetric group. A homomorphism is canonical when $n=k$ and belongs to the canonical construction in the paper; it is standard when $k\ne n$ and $k\mid n$ and belongs to the standard family; and it is derived from a canonical or standard homomorphism when it is obtained by the paper's derivation construction.

Classification conjecture. Every good non-cyclic transitive homomorphism $B_k\to S_n$ is canonical when $n=k$, standard when $k\ne n$ and $k\mid n$, or derived from a canonical or standard homomorphism when $k\nmid n$.

This is the paper's proposed complete classification of good transitive non-cyclic permutation representations. The paper proves representative cases but does not establish the full assertion.