Strong rigidity of periodic points outside equivariant torus quotients

Let $F$ be a field of characteristic zero and let $S_F$ be a normal affine surface over $F$. Suppose that $f,g\rbrace\in\operatorname{Aut}(S_F)$ are loxodromic automorphisms. Strong rigidity conjecture. The following are equivalent:

1. $\operatorname{Per}(f)\cap\operatorname{Per}(g)$ is Zariski dense.
2. There exist $N,M\in\mathbf{Z}\setminus\{0\}$ such that $f^N=g^M$.

This equivalence is conjectured for normal affine surfaces except when $S_F$ is an equivariant quotient of $\mathbb{G}_m^2$, since such quotients provide counterexamples to strong rigidity of periodic points.