The bi-automaticity conjecture for Shephard groups

A Shephard group is a group with a presentation consisting of Artin relations and finite-order relations on its generators; equivalently, for a finite generating set $X=\{x_1,\ldots,x_n\}$ and a symmetric matrix $\mathcal{C}=(m_{ij})$ with $m_{ij}\in\{2,3,4,\ldots\}\cup\{\infty\}$, it has the presentation

$$\left\langle X\mathrel{\middle|}\begin{array}{l} \left[x_i,x_j\,;\,m_{ij}\right]_{ar}=\left[x_j,x_i\,;\,m_{ji}\right]_{ar},\quad 1\leq i<j\leq n,\\ x_i^{m_{ii}}=1,\quad i=1,\ldots,n \end{array}\right\rangle.$$

Shephard groups' bi-automaticity conjecture. Every Shephard group is bi-automatic. The conjecture extends analogous conjectures for Artin and Coxeter groups. The paper proves it for an infinite family of Shephard groups that are neither Artin nor Coxeter groups, while the general case remains open.