Setup: Let $X$ be some probability space with probability measure $\mu$ (so $\mu(X)=1$). And let $G$ be a one real perameter topological group so $G=\{g_s:s\in\mathbb{R}\}$, and for the sake of simplicity let's assume the topology on $G$ corresponds to the standard topology on $\mathbb{R}$ with a homeomorphism given by $g_s\mapsto s$. Lastly we have a map $G\to MPCT(X)$ (space of continuous measure preserving transforms of $X$) where the resulting map $G\times X\to X$ is continuous in both perameters.
We say $G$ is mixing if for every sequence $\{g_n\}$ with $g_n$ eventually leaving any compact set we have $$\mu(g_n^{-1}(A)\cap B)\to\mu(A)\mu(B)$$ for any two measurable sets $A,B\subset X$. Now let $\langle g_s\rangle$ be the subgroup generated by an element $g_s\in G$. Here is my question, if we have that $\langle g_s\rangle$ is mixing for any generator $g_s\in G$, does that imply $G$ is mixing? It doesn't seem like it should necessarily but I can't construct a counter example.