Stability of nontrivial graph pairs

A graph pair $(\Gamma, \Sigma)$ is called stable if every automorphism of the direct product $\Gamma\times\Sigma$ is induced componentwise by automorphisms of $\Gamma$ and $\Sigma$. A graph is twin-free if no two distinct vertices share the same neighbourhood in the graph. Two graphs $\Gamma$ and $\Sigma$ are coprime with respect to the direct product if there is no graph $\Delta$ of order greater than $1$ such that $\Gamma\cong\Gamma'\times\Delta$ and $\Sigma\cong\Sigma'\times\Delta$ for some graphs $\Gamma'$ and $\Sigma'$. A graph pair $(\Gamma,\Sigma)$ is nontrivial if $\Gamma$ and $\Sigma$ are coprime connected twin-free graphs and exactly one of them is bipartite. In this paper, we prove that if $\Gamma$ is non-bipartite, stable, and factor-loopless, then each nontrivial graph pair $(\Gamma,\Sigma)$ is stable. This gives a partial answer to [Question~19, Qin, Xia and Zhou, Discrete Math., 113856, (2024)] and proves the factor-loopless case of [Conjecture~1.3, Wang, Qin and Xia, arXiv:2509.26170]. We also give affirmative answers to [Questions~3.5, 3.6, Gan, Liu and Xia, J. Combin. Theory Ser. B, 140--164, (2025)] and a negative answer to [Question~3.7, Gan, Liu and Xia, J. Combin. Theory Ser. B, 140--164, (2025)].