Normal-subgroup conjecture for non-trivial braid closures

Let $B_n$ be the braid group on $n$ strands. For a braid, its closure is the link obtained by joining corresponding endpoints. A subgroup $H$ of $B_n$ is non-trivial normal if it is normal and contains an element other than the identity braid. Normal-subgroup conjecture. Let $H$ be a non-trivial normal subgroup of $B_n$. Then there exists $\beta_1\in H$ and $\beta_2\in B_n$ such that the closure of $\beta_2$ is the unknot but the closure of $\beta_1\beta_2$ is a non-trivial knot. This is presented as a weakening of the preceding powers conjecture, allowing products of conjugates and arbitrary braids whose closures are the unknot; the supplied status evidence indicates that the claim has been disproved.