Dehornoy's class bound for solutions with abelian permutation group

Let $(X,r)$ be an involutive solution of size $n$, and let $g(n)$ denote the bound used in the surrounding text for solutions whose permutation group is abelian. Abelian permutation-group conjecture. The Dehornoy's class of $(X,r)$ with abelian permutation group is bounded by $g(n)$. The statement is posed as a natural question in the source, and no resolution is supplied.