Query-complexity conjecture for amenable group presentations

Let $S$ be a finite generating set and let $E$ be a finite set of relators such that

$$\Gamma=\langle S\mid E\rangle$$

is a finitely presented amenable group. Let the Følner function of $\Gamma$ measure the sizes of Følner sets needed to achieve prescribed invariance. Query-complexity conjecture. The relator set $E$ is testable with query complexity bounded above by a function of $|S|$, $\sum_{w\in E}|w|$, and the Følner function of $\Gamma$. This conjecture concerns quantitative testability for amenable group presentations; the supplied source presents it as an open conjecture and gives no resolution.