Full-twist conjecture for odd-superbridge torus braids

For integers $n\ge3$ and $k\ge0$, consider the closure of the $n$-braid

$$\sigma_{1}^{2k}(\sigma_1\sigma_2\cdots\sigma_{n-1})^{2n-1}.$$

A knot with superbridge index $m$ is called an $m$-superbridge knot.

Full-twist odd-superbridge conjecture. For $n\ge3$ and $k\ge0$, the closure of this braid is a $(2n-1)$-superbridge knot.

The case $k=0$ is the torus knot of type $(n,2n-1)$, whose superbridge index is $2n-1$. The conjecture proposes that inserting any nonnegative even power of the first braid generator preserves that superbridge index, and is intended to provide infinitely many odd-superbridge knots.