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Equi-Energy Sampling (EES, for short) is a method to speed up the convergenceof the Metropolis chain, when the latter is slow. We show that there are stillmodels like the mean-field Potts model, where EES does not converge rapidly incertain temperature regimes. Indeed we will show that EES is slowly mixing onthe mean-field Potts model, in a regime below the critical temperature. Thoughwe will concentrate on the Potts model with three colors, our arguments remainvalid for any number of colors $q \ge 3$, if we adapt the temperature regime.For the situation of the mean-field Potts model this answers a question posedin \cite{HuaKou}.

Equi-Energy sampling does not converge rapidly on the mean-field Potts model with three colors close to the critical temperature