Illustration of the fundamental insight motivating this work. Eccentric binaries radiate GWs at the characteristic frequency associated to pericenter passage. Therefore, they produce GWs at high frequencies (in particular at the characteristic “peak” frequency) over a larger portion of their entire evolution. They enter the sensitive frequency band of ground-based GW detectors at a wider separation, at which EEs are strong with respect to relativistic effects. Therefore, we expect a stronger trace of binary environments in the GW signal of eccentric sources.