Verify Condition B for the exceptional covering of type B3

Prove or disprove that the $[?]$-regular covering of the spherical type $B_3$ Artin–Tits monoid with $S_1=\{\sigma_0\}$ and $S_2=\{\sigma_1,\sigma_2\}$ satisfies Condition B.

Background

The specified covering of the type B3B_3 Artin–Tits monoid is [?][?]-regular and satisfies Condition A by the paper’s main result. Neither the authors’ theoretical results nor their computational experiments produced a counterexample to Condition B, but they also did not establish Condition B.

Resolving this case would determine whether the covering gives a new example relevant to the construction of Dehornoy structures and left-orderings.

References

We were unable to obtain a counterexample by hand of Condition B, nor a proof that Condition B is satisfied. However, we suspect that the latter is true.

— Ordering, Artin--Tits group, Garside structure  (2609.04757 - Fromentin et al., 4 Sep 2026) in Section 1, Introduction, subsection “What is done in this paper”