Finite determination conjecture for cubical coarse median structures on right-angled Coxeter groups

Let $W_{\Gamma}$ be a right-angled Coxeter group. A cubical coarse median rigidity conjecture.

1. There are only finitely many cubical coarse median structures on $W_{\Gamma}$.
2. Each cubical coarse median structure on $W_{\Gamma}$ is completely determined by its restriction to maximal parabolic virtually-abelian subgroups.

The conjecture proposes a weaker form of coarse cubical rigidity for all right-angled Coxeter groups. The stronger rigidity statement is known to fail for some one-ended, irreducible examples, while the claims above were not established in the source.