Quasi-isometry invariance of higher-dimensional homological Dehn functions

Let $G$ and $H$ be groups of type $FP_{n+1}(R)$, and let $f_n^G$ and $f_n^H$ denote their $n$-dimensional homological Dehn functions over $R$. Suppose that $G$ and $H$ are quasi-isometric.

Quasi-isometry invariance conjecture. Then

$$f_n^G\approx f_n^H.$$

The preceding results establish the analogous invariance when the groups are of type $FH_n(R)$. Since $FP_2(R)$ is equivalent to $FH_2(R)$ but the equivalence is unknown for $n>2$, this conjecture addresses whether the homological Dehn function remains a quasi-isometry invariant under the weaker finiteness hypothesis $FP_{n+1}(R)$.