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In classical probability theory, the probability of the conjunction of two events is smaller than the probability of only one of these events. Yet, agents do not always empirically judge in this way: this is the traditional conjunction fallacy. One of the currently promising accounts of this paradox relies on so-called quantum-like models, which have been developed from mathematical tools used in quantum theory. But are these models empirically adequate? Which versions of these models can be used? In particular, can the simplest versions, the non-degenerate ones, be sufficient? We propose here an original experimental protocol to test the quantum-like models for the conjunction fallacy in the lab. The results we obtain suggest that the non-degenerate models are not empirically adequate, and that future research on quantum-like models should consider degenerate ones. Classification JEL : C60, C91, D03.