The conjectural braid-group presentation for the complex reflection group $G_{31}$

Let $G_{31}$ 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 $G_{31}$ braid-group conjecture. The braid group associated with $G_{31}$ admits the presentation

$$\left\langle s,t,u,v,w \left| \begin{array}{c} sts=tst,\ tut=utu,\ uvu=vuv,\ vwv=wvw \\ sv=vs,\ tv=vt,\ tw=wt \\ suw=uws=wsu \end{array} \right. \right\rangle.$$

These relations imply that $(stuwv)^6$ is central.

The source describes this as confirming a conjectural presentation of Broué, Malle, and Rouquier, with computational evidence behind the proposed presentation. The group is four-dimensional but requires at least five generating reflections.