Coarse preservation of maximal product subgraphs

Establish whether every quasi-isometry between quasi-median graphs on which groups act geometrically and specially sends each maximal product subgraph of the source within uniformly bounded Hausdorff distance of a maximal product subgraph of the target.

Background

Theorem 2.10 in the paper proves the relevant coarse preservation statement for median graphs associated to right-angled Artin groups, and another cited result covers median graphs of cubical dimension two. The authors ask for a broader result for groups acting geometrically and specially on quasi-median or median graphs.

References

Given a quasi-isometry \eta : X_1 \to X_2, does there exist a constant C \geq 0 such that \eta sends every maximal product subgraph of X_1 at Hausdorff distance \leq C from some maximal product subgraph of X_2?

— Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups  (2503.08411 - Abbott et al., 11 Mar 2025) in Section 6, paragraph “Right-angled Coxeter groups,” Question