  We prove that the presentations $\langle x,y | [x,y],1 \rangle$ and $\langle
x,y | [x,[x,y^{-1}]]^2y[y^{-1},x]y^{-1},[x,[[y^{-1},x],x]] \rangle$ are not
$Q^*$-equivalent even though their standard complexes have the same simple
homotopy type.
