Tame reduction to the trivial multifraction for Artin-Tits monoids

Let $M$ be an Artin–Tits monoid, let $\mathcal{F}_{M}$ be the set of multifractions over $M$, and let $\operatorname{red}_{\mathrm{t}}$ denote tame reduction. A multifraction is unital if it represents the identity in the enveloping group of $M$, and $\underline{1}$ denotes the trivial multifraction. Tame-reduction conjecture. For every unital multifraction $\underline{a}\in\mathcal{F}_{M}$, one has

$$\operatorname{red}_{\mathrm{t}}(\underline{a})=\underline{1}.$$

This conjecture concerns a weaker, restricted reduction procedure. The supplied text reports that no counterexample had been found for Artin–Tits monoids, but the conjecture is not marked as resolved.