K-theory equality for B^*_r(G,H) with co-amenable or amenable subgroups

Ascertain whether, for a subgroup H of G that is co-amenable (respectively amenable), the Banach *-algebra B^*_r(G,H) has the same K-theory as the group Banach algebra ℓ^1(G) (respectively the reduced group C*-algebra C^*_r(G)).

Background

Earlier in the paper, the authors relate co-amenability of H in G to norm equalities and growth properties of the Schreier graph. They also show several isoradial and relatively spectral embeddings under growth assumptions.

This question asks whether those analytic properties extend to K-theory isomorphisms between B^*_r(G,H) and classical group algebras in the presence of co-amenability or amenability.