Sharp universal bound for Dehornoy's class

Let $(X,r)$ be an involutive solution of size $n$, and define

$$a_n:=\max\left\{\prod_{i=1}^k n_i\mathrel{\middle|}k\in\mathbb{N},\ 1\leq n_1<\cdots<n_k,\ n_1+\cdots+n_k=n\right\}.$$

Dehornoy's class bound conjecture. The Dehornoy's class is bounded by $a_n$, and the bound $a_n$ is minimal. The claim is stated as a conjecture in the cited literature; its resolution is not indicated in the supplied text.