The conjectural braid-group presentation for the complex reflection group $G_{34}$

Let $G_{34}$ be the complex reflection group, and let $s,t,u,v,w,x$ denote the generators in the proposed braid-group presentation. The phrase “relations of $G_{33}$” refers to the relations in the preceding proposed presentation for $G_{33}$, and the notation $\langle s,t,u,v,w,x\mid R\rangle$ denotes the group generated by these elements subject to the displayed relations.

The $G_{34}$ braid-group conjecture. The braid group associated with $G_{34}$ admits the presentation

$$\left\langle s,t,u,v,w,x \left| \begin{array}{c} \text{relations of }G_{33}\text{ +} \\ xvx=vxv,\ xs=sx,\ xt=tx,\ xv=vx,\ xw=wx \end{array} \right. \right\rangle.$$

These relations imply that $(stuvwx)^7$ is central.

The proposed presentation extends the preceding one for $G_{33}$ by adjoining the generator $x$ and the displayed relations. The source presents it as conjectural and gives no resolution beyond the stated structural relation.