Transferring K-theory equality from H to G under hybrid rapid decay

Ascertain whether, for a pair of groups (G,H) that has the hybrid rapid decay property and for which 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.

Background

The authors prove that hybrid rapid decay yields spectral and K-theoretic consequences for completions associated with (G,H). They ask whether K-theory equality at the subgroup level (for ℓ^1(H) and C^*_r(H)) lifts to the ambient pair, producing equality between B^*_r(G,H) and C^*_r(G).

A positive answer would extend known results for amenable groups to broader settings controlled by hybrid rapid decay.