Bader–Kropholler–Vankov's quasi-isometry invariance conjecture for homological filling functions

Let $G$ and $H$ be quasi-isometric groups of type $\FP_n(R)$, where $R$ is an arbitrary coefficient ring. Write $\operatorname{FV}_{R,G}^{n}$ and $\operatorname{FV}_{R,H}^{n}$ for their homological filling functions with coefficients in $R$, and let $\approx$ denote the usual equivalence of filling functions. Bader–Kropholler–Vankov's conjecture. Then

$$\operatorname{FV}_{R,G}^{n} \approx \operatorname{FV}_{R,H}^{n}.$$

This extends the known quasi-isometry invariance result for groups of type $\mathrm{FH}_n(R)$ to the broader class of groups of type $\FP_n(R)$. The equivalence of these finiteness conditions is itself a well-known open problem for $n>2$, and the conjectured invariance remains open.