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The post-merger gravitational wave (GW) emission from a binary neutron star merger is expected to provide exciting new constraints on the dense-matter equation of state (EoS). Such constraints rely, by and large, on the existence of quasi-universal relations, which relate the peak frequencies of the post-merger GW spectrum to properties of the neutron star structure in a model-independent way. In this work, we report on violations of existing quasi-universal relations between the peak spectral frequency, f_2, and the stellar radius, for EoSs models with backwards-bending slopes in their mass-radius relations (such that the radius increases at high masses). The violations are extreme, with variations in f_2 of up to ~600 Hz between EoSs that predict the same radius for a 1.4 Msun neutron star, but that have significantly different radii at higher masses. Quasi-universality can be restored by adding in a second parameter to the fitting formulae that depends on the slope of the mass-radius curve. We further find strong evidence that quasi-universality is better maintained for the radii of very massive stars (with masses 2 Msun). Both statements imply that f_2 is mainly sensitive to the high-density EoS. Combined with observations of the binary neutron star inspiral, these generalized quasi-universal relations can be used to simultaneously infer the characteristic radius and slope of the neutron star mass-radius relation.