The conjectural braid-group presentation for the complex reflection group $G_{33}$

Let $G_{33}$ be the complex reflection group, and let $s,t,u,v,w$ denote the generators in the proposed braid-group presentation. The notation $\langle s,t,u,v,w\mid R\rangle$ denotes the group generated by these elements subject to the displayed relations.

The $G_{33}$ braid-group conjecture. The braid group associated with $G_{33}$ admits the presentation

$$\left\langle s,t,u,v,w \left| \begin{array}{c} sts=tst,\ tut=utu,\ uvu=vuv,\ wtw=twt,\ wuw=uwu \\ su=us,\ sv=vs,\ tv=vt,\ ws=sw,\ wv=vw \\ tuwtuw=uwtuwt=wtuwtu \end{array} \right. \right\rangle.$$

These relations imply that $(stuvw)^9$ is central.

The displayed relations do not coincide with the homogeneous part of the Broué–Malle–Rouquier presentation for $G_{33}$, although the relations involving $t,u,w$ coincide with those for the braid group of $G(3,3,3)$. One of the three displayed relations is redundant.