a finite system. It is known [@MeMo2000; @FeCa2007; @PeIbLo2008] that below but close to $T_c$ the system quickly relaxes to a disordered (paramagnetic) metastable state, with a life time that diverges as the quench temperature $T$ approaches $T_c$ (see for example Fig.4 in Ref.[@LoFeGrCa2009]). This state is indistinguishable from one in equilibrium in the sense of local dynamics, namely, two times correlations depends only on the difference of times, while one time averages are stationary [@PeIbLo2008].

Nevertheless, the existence of metastability in the thermodynamic limit is still an open problem [@PeIbLo2008]. In Ref.[@Bi1981] Binder studied static and dynamic critical behavior of the model (\[Hamiltonian\]) for $q=3,4,5,6$. Using standard Monte Carlo procedures he obtained good agreement with exact results for energy and free energy at the critical point and critical exponents estimates for $q=3$ in agreement with high-temperature series extrapolations and real space renormalization-group methods. When analyzing the $q=5$ and $6$ cases he realized that the transition is, in fact, a very weak first order transition, where pronounced “pseudocritical” phenomena occur. He studied system sizes from $N=16\times16$ up to $N=200\times200$, and observation times up to $10^4 \mathit{MCS}$ (a Monte Carlo Step $\mathit{MCS}$ is defined as as a complete cycle of $N$ 