Untangling trigonal diagrams

Let $K$ be a link of Conway’s normal form $C(m)$, $m\ge 0$, or $C(m,n)$ with $mn\text{textgreater}0$, and let $D$ be a trigonal diagram of $K\mathrm{.}$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.