Let $G$ be a finite soluble group, let $x$ be a non-identity element of $G$, let $v$ be an $n$-tuple in $G$, and let $w$ be a word in $n+2$ variables. Does there exist $(G,x,v,w)$ satisfying the following equations?

$w(1,1,v)=w(x,1,v)=w(1,x,v)=1; w(x,x,v)=x.$

As for why the soluble condition is there: let $w(a,b,c) = [a,c^{-1}bc]$, and suppose we have chosen $x,v \in G$ such that $[x,v^{-1}xv]=x$. This seems quite a plausible equation in a simple group (at first glance, anyway) but obviously has no non-trivial solutions in a soluble group, because the left-hand side is further down the derived series than the right-hand side. But perhaps a more complicated word does not have this limitation.