In this paper we study the relation between the property of detailed balance and the ability of discriminating between different ligands for a class of stochastic models of kinetic proofreading. We prove the existence of a critical amount of lack of detailed balance that the kinetic proofreading models must have in order to have strong specificity for a value of the binding energy $σ$. We also prove that the fact that a kinetic proofreading model has a lack of detailed balance that is larger than the critical one does not necessarily yield strong discrimination properties. Indeed, there exist different sets of chemical rates, leading to the same amount of lack of detailed balance, that have strong discrimination property in some cases and not in others.