Transferring K-theory equality from H to G without hybrid rapid decay

Ascertain whether, for a subgroup H<G such that ℓ^1(H) and C^*_r(H) have the same K-theory, it follows that B^*_r(G,H) and C^*_r(G) have the same K-theory, without assuming the hybrid rapid decay property for the pair (G,H).

Background

This question removes the hybrid rapid decay assumption from the previous K-theory transfer problem. It probes whether subgroup-level K-theory equality can imply ambient equality for B^*_r(G,H) and C^*_r(G) in general.

Such a result would provide a broad K-theoretic bridge from subgroup algebras to ambient group algebras, beyond the analytic control afforded by hybrid rapid decay.