Property H conjecture for Artin–Tits presentations

Let $(S,R)$ be a positive presentation, let $\mathcal{W}(S)$ be its words, and write $\langle S\mid R\rangle$ for the presented group. A positive presentation $(S,R)$ satisfies Property $H$ if every word $w\in\mathcal{W}(S)$ representing $1$ in $\langle S\mid R\rangle$ satisfies

$$w \mathrel{\mathrel{\overset{\scriptscriptstyle 0,1,2}{\rightsquigarrow}}_{\hspace{-0.9ex}{\scriptscriptstyle R}}} \varepsilon.$$

Property H conjecture. Every Artin–Tits presentation satisfies Property $H$. The conjecture asks whether transformations of type $\infty$ are unnecessary for reducing every word representing the identity to the empty word. It is the central question addressed by the paper; the supplied source does not indicate whether it has been resolved.