Stability conjecture for nontrivial direct product graph pairs

A graph pair $(\Gamma,\Sigma)$ is nontrivial if $\Gamma$ and $\Sigma$ are coprime connected twin-free graphs and exactly one of them is bipartite. A graph pair is stable if it has no unexpected automorphisms under the direct product construction. Stability conjecture. Let $(\Gamma,\Sigma)$ be a nontrivial graph pair with $\Sigma$ bipartite. Then $(\Gamma,\Sigma)$ is stable if and only if $\Gamma$ is stable.

The conjecture would eliminate the need for any additional condition in the reduction of stability for a nontrivial pair with bipartite second factor to stability of the single graph $\Gamma$. Explicitly determining stable nontrivial pairs is difficult, even when $\Sigma=K_2$, so the conjecture represents a broad proposed reduction of the stability problem.