Entropic forces tend to demix polymers in confinement, which has been argued to at least facilitate DNA segregation in cylindrical bacteria. Ring polymers as found in modem bacteria such as Escherichia coli experience even stronger segregating forces than linear ones due to the fact that rings additionally constrain themselves. Using a territorial "renormalized" Flory approach we obtain a scaling prediction for the segregation force and speed of ring polymers and confirm this prediction by molecular dynamics simulations. The ring topology also affects the induction phase, when the initial symmetry is broken before segregation sets in We show that the induction time still scales exponentially with the chain length and thus dominates the overall time scale of entropic segregation, although it is significantly shorter than the one for linear chains.