The conjectural braid-group presentation for the complex reflection group $G_{29}$

Let $G_{29}$ be the complex reflection group, and let $s,t,u,v$ denote the generators in the proposed braid-group presentation. The notation $\langle s,t,u,v\mid R\rangle$ denotes the group generated by these elements subject to the displayed relations.

The $G_{29}$ braid-group conjecture. The braid group associated with $G_{29}$ admits the presentation

$$\left\langle s,t,u,v \left| \begin{array}{c} sts=tst,\ tut=utu,\ uvu=vuv,\ tvtv=vtvt, \\ su=us,\ sv=vs,\ utvutv=tvutvu \end{array} \right. \right\rangle.$$

These relations imply that $(stuv)^5$ is central.

The presentation was already given by Broué, Malle, and Rouquier, but its interpretation as a braid-group presentation was not originally conjectured; the authors report computational evidence supporting it.