Commensurability closure of the CRn classes

Ascertain whether, for each n, the class CRn of groups satisfying the algebraic cheap n-rebuilding property is closed under commensurability; equivalently, determine whether membership in CRn is preserved when passing between commensurable groups.

Background

The algebraic cheap rebuilding property CRn is formulated using residual chains, which makes induction arguments straightforward but complicates restriction to finite index subgroups (axiom (B-res)).

Because (B-res) is unknown for CRn, standard arguments that imply closure under commensurability are not available. The geometric analogue in Abért–Bergeron–Frączyk–Gekhtman is known to be commensurability-invariant, and the authors suggest a strengthened algebraic framework (via Farber neighborhoods) might achieve the same.