Cancellativity conjecture for the monoids \mathcal{H}_n^+ and \Sigma_n

Let $n\geq 3$. The monoid $\mathcal{H}_n^+$ is defined by the presentation

$$\left\langle \rho_1,\rho_2,\dots,\rho_n\ \middle|\ \rho_1\rho_j\rho_i=\rho_{i+1}\rho_j\text{ for }1\leq i<j\leq n\right\rangle.$$

Let $\mathcal{B}_{n+1}$ be the braid group on $n+1$ strands, with standard generators $\sigma_1,\dots,\sigma_n$, and let $\Sigma_n$ be the submonoid generated by $\sigma_1,\sigma_1\sigma_2,\dots,\sigma_1\sigma_2\cdots\sigma_n$. Cancellativity and embedding conjecture. The monoid $\mathcal{H}_n^+$ is cancellative and isomorphic to $\Sigma_n$ via

$$\rho_i\mapsto\sigma_1\sigma_2\cdots\sigma_i.$$

In particular, it embeds into $\mathcal{B}_{n+1}$, which is therefore isomorphic to its group of fractions. The conjecture would also imply that $\Sigma_n$ admits a finite presentation. The case $n=3$ was previously conjectured by Dehornoy; the general statement remains open.