Cross-confluence of reduction for Artin-Tits monoids

Let $M$ be an Artin–Tits monoid and let $\mathcal{R}_{M}$ be its reduction system on multifractions. Two multifractions are right reducts of the same multifraction if they are obtained from it by sequences of right reductions; the system is cross-confluent when any two such reducts admit a common left reduct. Cross-confluence conjecture. For every Artin–Tits monoid $M$, the system $\mathcal{R}_{M}$ is cross-confluent.

Cross-confluence would imply semi-convergence and thus provide a route to the word problem without requiring ordinary convergence. The paper presents it as its main conjecture, supported by examples and computations, but no general proof is given.