Conjecture A on semi-convergence for Artin–Tits monoids

Let $M$ be an Artin–Tits monoid, and let $\mathcal{R}$-reduction be the reduction system on multifractions associated with $M$. Conjecture A. $\mathcal{R}$-reduction is semi-convergent for every Artin–Tits monoid. The conjecture is known for every Artin–Tits monoid of FC type, while the remaining cases are open.