Non-bonded squares and uniqueness of cubical coarse medians

Prove or disprove that the existence of a non-bonded induced square in a defining graph implies that the associated right-angled Coxeter group has more than one cubical coarse median structure.

Background

The paper uses a theorem of Fioravanti, Levcovitz, and Sageev stating that if every induced square in a defining graph is bonded, then the associated RACG has coarse cubical rigidity.

Those authors further suggested the converse: a non-bonded square should yield non-uniqueness. The converse is explicitly unresolved, and the present paper uses random graph methods only to establish a threshold for the disappearance of non-bonded squares, not to prove the converse implication.

References

They further suggested that this should be an if and only if, i.e., that the existence of non-bonded squares implies non-uniqueness of the coarse median structure on $W_{\Gamma}$, though they were unable to prove this.

— Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups  (2502.18165 - Behrstock et al., 25 Feb 2025) in Section 5, subsection on implications for the geometry of the random RACG