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

Let $\mathrm{\Sigma }$ be a compact surface of genus $g\ge 1$ 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 $g\ge 2$ (our argument provides an alternative, very simple construction compared to previous publications) while the case $g=1$ is new.