  Equi-Energy Sampling (EES, for short) is a method to speed up the convergence
of the Metropolis chain, when the latter is slow. We show that there are still
models like the mean-field Potts model, where EES does not converge rapidly in
certain temperature regimes. Indeed we will show that EES is slowly mixing on
the mean-field Potts model, in a regime below the critical temperature. Though
we will concentrate on the Potts model with three colors, our arguments remain
valid 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 posed
in \cite{HuaKou}.
