Margalit's conjecture on the smallest finite non-cyclic quotient of the braid group

Let $B_n$ be the braid group on $n$ strands, and let $\Sigma_n$ denote the finite quotient under consideration. A finite quotient is non-cyclic if it is not a cyclic group. Margalit's conjecture. For $n \geq 5$, $\Sigma_n$ is the smallest finite non-cyclic quotient of $B_n$. This conjecture concerns the problem of determining the smallest finite quotients of braid groups that are not cyclic. The preceding theorem gives a lower bound for the size of every non-cyclic finite quotient, but the conjectured minimality of $\Sigma_n$ remains unresolved here.