It is known that there are non-amenable groups not containing $F_2$, the free group on two generators. We can even have that every 2-generated subgroup is finite.

But is there a non-amenable group $G$ where for some $n$, $G$ is length-$n$ unfree in the following sense?

Definition. $G$ is length-$n$ unfree if for all $a,b\in G$, there exist words $u\ne v$ of length $n$ over the alphabet $\{a,b\}$ such that $u=v$ in $G$.