Failure of automorphism invariance for Andrews–Curtis equivalence

Let $F_2$ be the free group on $x,y$, let ${\mathcal B}_2$ be the set of balanced presentations of the trivial group with two generators, and let $\operatorname{Aut}(F_2)$ act componentwise on pairs. Write $(u,v)\sim_{\operatorname{AC}}(u',v')$ when the pairs are related by Andrews–Curtis moves. Failure of automorphism invariance. It is not true that for every $(u,v)\in{\mathcal B}_2$ and every $\varphi\in\operatorname{Aut}(F_2)$, one has $(u,v)\sim_{\operatorname{AC}}(\varphi(u),\varphi(v))$. This contradicts the possibility that adding automorphism moves never enlarges Andrews–Curtis orbits for all balanced presentations. The source gives no resolution evidence beyond the asserted negative statement.