A conjunction fallacy happens when an agent judges that a conjunction – for instance, « Linda is a feminist and a bank teller » – is more likely than one of the components – for instance, « Linda is a bank teller ». To account for this fallacy, quantum-like models have been recently proposed. In this paper, we show that these quantum-like models imply some theoretical predictions (originally derived in Boyer-Kassem, Duchêne and Guerci, 2015). To test them, we design and run a new Linda-like experiment. The results show that non-degenerate quantum-like models like the ones discussed in the literature are not suitable. We take it to suggest that degenerate quantum-like models should be the focus of forthcoming research in the area.