Gravitational waves from binary neutron star (BNS) mergers can constrain nuclear models, predicting their equation of state (EOS). Matter effects on the inspiral-merger signal are encoded in the multipolar tidal polarizability parameters, whose leading order combination is sufficient to capture, with high accuracy, the key features of the merger waveform. Similar EOS-insensitive relations exist for the post-merger signal and can be used to model the emissions from the remnant. Several works suggested that the appearance of new degrees of freedom in high-density post-merger matter can be inferred by observing a violation of these EOS-insensitive relations. Here, we demonstrate a Bayesian method to test such an EOS-insensitive relation between the tidal polarizability parameters (or any other equivalent parameter) and the dominant post-merger frequency using information from the pre-and-post-merger signal. Technically, the method is similar to the inspiral-merger-ringdown consistency tests of General Relativity with binary black holes. However, differently from the latter, BNS pre/post-merger consistency tests are conceptually less informative and they only address the consistency of the assumed EOS-insensitive relation. Specifically, we discuss how such tests cannot conclusively discriminate between an EOS without respecting such a relation and the appearance of new degrees of freedom (or phase transitions) in high-density matter.