Uniform 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. The system is uniformly cross-confluent if there is a map $\nabla$ from multifractions over $M$ to themselves such that every right reduct of a multifraction $\underline{a}$ reduces to $\nabla(\underline{a})$. Uniform cross-confluence conjecture. For every Artin–Tits monoid $M$, the system $\mathcal{R}_{M}$ is uniformly cross-confluent.

Uniform cross-confluence strengthens cross-confluence by requiring a common target depending only on the starting multifraction. It holds when reduction is convergent, but its validity for arbitrary Artin–Tits monoids is left open.