Let $(G,\cdot)$ be a group and $\phi:G\rightarrow\mathbb R$ bounded. Let me say that the pair $(G,\phi)$ is amenable if there is a finitely additive probability measure $\mu$ on $G$ such that for all $y\in G$

$$ \int \phi(x)d\mu(x)=\int \phi(x\cdot y)d\mu(x)=\int\phi(y\cdot x)d\mu(x) $$

Question:Does there exist a non-amenable group such that the pair $(G,\phi)$ is amenable for all $\phi\in\ell^\infty(G)$?