Effect of passing to the commensurated core on the quasi-isometry type of \mathcal{K}(G,P)

Ascertain, for general group pairs (G, P), how replacing the peripheral collection P by a commensurated core Q affects the quasi-isometry type of the coset intersection complex \mathcal{K}(G,P); in particular, determine conditions under which the quasi-isometry type is preserved when passing from P to Q.

Background

The paper defines the commensurated core of a group pair using setwise stabilizers of maximal simplices of the coset intersection complex and shows that, for certain right-angled Artin groups, replacing the maximal standard abelian subgroups P by the core Q preserves the quasi-isometry type of \mathcal{K}(G,P).

Beyond these cases, the authors state that the general impact of this replacement on the quasi-isometry type is not understood, highlighting a broader problem of identifying when the quasi-isometry type is invariant under passage to the commensurated core.