Cocompactness from finite dimensionality of the coset intersection complex

Determine whether, for every group pair (G, P), the action of G on the coset intersection complex \mathcal{K}(G,P) is cocompact whenever \mathcal{K}(G,P) is finite dimensional; equivalently, ascertain whether there exists a group pair (G, P) of finite height that does not admit a finite commensurated core.

Background

The authors define finite packing and introduce the commensurated core of a group pair, proving that finite packing is equivalent to having both finite height and a finite commensurated core. This connects geometric finiteness of the coset intersection complex to subgroup-structural data.

They note that infinite-dimensionality of the complex rules out cocompactness, but the converse direction—whether finite dimensionality forces cocompactness (equivalently, finite height forces a finite commensurated core)—is not known. They explicitly pose this as a question following the remark.