Arouche's subgroup-index consequence for braid commutator subgroups

Let $B_n$ be the braid group on $n$ strands and let $K_n=[B_n,B_n]$ be its commutator subgroup. A subgroup has index $r$ if its coset action gives a transitive representation $K_n\to S_r$. Arouche's subgroup-index consequence. For $n\geq 5$ and $2\leq r\leq n-1$, there are no subgroups of $K_n$ with index $r$. Moreover, every nontrivial representation $\rho$ of $K_n$ into $S_n$ is transitive. The source presents this as a consequence of the preceding triviality conjecture, so its resolution is likewise not established in the supplied text.