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It was recently proposed that there exists a "gateway" in the orbital parameter space through which Centaurs transition to Jupiter-family Comets (JFCs). Further studies have implied that the majority of objects that eventually evolve into JFCs should leave the Centaur population through this gateway. This may be naively interpreted as gateway Centaurs being pristine progenitors of JFCs. This is the point we want to address in this work. We show that the opposite is true: gateway Centaurs are, on average, more thermally processed than the rest of the population of Centaurs crossing Jupiter's orbit. Using a dynamically-validated JFC population, we find that only $\sim 20\%$ of Centaurs pass through the gateway prior to becoming JFCs, in accordance with previous studies. We show that more than half of JFC dynamical clones entering the gateway for the first time have already been JFCs -they simply avoided the gateway on their first pass into the inner solar system. By coupling a thermal evolution model to the orbital evolution of JFC dynamical clones, we find a higher than 50\% chance that the layer currently contributing to the observed activity of gateway objects has been physically and chemically altered, due to previously sustained thermal processing. We further illustrate this effect by examining dynamical clones that match the present-day orbits of 29P/Schwassmann-Wachmann 1, P/2019 LD2 (ATLAS), and P/2008 CL94 (Lemmon).