The origin and composition of ultra-high-energy cosmic rays (UHECRs) remain a mystery. The proton dip model describes the shape of the cosmic ray spectrum above $10^9$ GeV by the effect of a pure proton spectrum propagating through the cosmic microwave background. In these interactions secondary neutrinos are produced.

We fit the recent UHECR spectrum measurements from the Telescope Array experiment under the assumption of pure proton composition, as assumed by the proton dip model.

We present a a full scan of the three main physical model parameters of UHECR-injection: source redshift evolution, injected maximal proton energy and spectral power-law index. We discuss how the result qualitatively changes compared to earlier two-parameter fits in the literature: a mild preference for a maximal energy cutoff at the sources instead of the Greisen-Zatsepin-Kuzmin (GZK) cutoff, hard injection spectra, and strong source evolution.

We show that the predicted neutrino flux exceeds the IceCube limit for any parameter combination. As a result, the proton dip model is challenged at more than $99\%$ C.L. This is strong evidence against the dip model independent of mass composition measurements.