Conjecture on knot semigroups of tripletwist knots

Let $l,m,n\geq 1$ be integers, and let $C(m,l,n)$ denote the indicated tripletwist knot diagram. Let $M(C(m,l,n))$ be its knot semigroup, and let ${\rm AS}(\mathbb{Z}_{(ml+1)n+m},S)$ denote the alternating sum semigroup on the cyclic group $\mathbb{Z}_{(ml+1)n+m}$ with generating subset $S$. Define

$$S=\displaystyle\bigcup_{i=0}^{n+1}\{i\}\cup\displaystyle\bigcup_{j=0}^{l+1}\{j n+1\}\cup\displaystyle\bigcup_{k=0}^{m-1}\{(kl+1)n+k\}.$$

Tripletwist knot-semigroup conjecture. If $(ml+1)n+m$ is odd, then

$$M(C(m,l,n))\simeq {\rm AS}(\mathbb{Z}_{(ml+1)n+m},S).$$

This extends the alternating-sum-semigroup pattern proved in the paper for double twist knots. No resolution of this tripletwist claim is given in the supplied text.