Schwartz (Soc Choice Welfare 3:271–291, 1990) proposed a solution concept (the Tournament Equilibrium Set) for choosing from a tournament and stated some conjectures about this solution. Laffond et al. (Math Sci Hum 123:37–44, 1993) introduced further conjectures and showed the links between some of them. In this short note, we show that one of the conjectures stated by Schwartz (1990) is false. We also complete a result given in Laffond et al. (1993).