Hofer's distance between eggbeaters and autonomous Hamiltonian diffeomorphisms on surfaces

Let $\Sigma$ be a compact surface of genus $g \geq 1$ equipped with an area form. We construct eggbeater Hamiltonian diffeomorphisms which lie arbitrarily far in the Hofer metric from the set of autonomous Hamiltonians. This result is already known for $g \geq 2$ (our argument provides an alternative, very simple construction compared to previous publications) while the case $g = 1$ is new.