Powers of a braid conjecture for producing non-trivial knot closures

Let $B_n$ be the braid group on $n$ strands, with standard generators $\sigma_1,\sigma_2,\dots,\sigma_{n-1}$. For a braid, its closure is the link obtained by joining corresponding endpoints. A braid is non-trivial if it is not the identity braid. Powers of a braid conjecture. Let $\beta$ be a non-trivial braid in $B_n$. There exists some integer $k$ such that the closure of $\beta^k\sigma_1\sigma_2\dots\sigma_{n-1}$ is a non-trivial knot. The conjecture is proposed because this closure always has the same Jones polynomial as the unknot when $\beta$ lies in the kernel of the Temperley–Lieb representation; the source states that it would imply the preceding conjecture, but does not establish it.