Conjecture: Quasi-isometry invariance of the n-isoperimetric class for FP_{n+1}(R) groups

Prove that for any normed ring R and integer n ≥ 1, if groups G and H are quasi-isometric and both are of type FP_{n+1}(R), then their n-homological isoperimetric classes over R are equivalent (f_n^G ≈ f_n^H).

Background

The paper establishes quasi-isometry invariance of the n-isoperimetric class under stronger geometric assumptions (type FH_{n+1}(R)) provided finiteness holds. However, for n > 2 the equivalence of FP_n(R) and FH_n(R) is unknown, and it is not known whether type FH_n(R) is quasi-isometry invariant.

Motivated by these obstructions, the authors conjecture that quasi-isometry invariance of the n-isoperimetric class should hold already at the purely algebraic finiteness level FP_{n+1}(R), bypassing the need to assume FH_{n+1}(R).