  The origin and composition of ultra-high-energy cosmic rays (UHECRs) remain a
mystery. The proton dip model describes their spectral shape in the energy
range above $10^9$ GeV by pair production and photohadronic interactions with
the cosmic microwave background. The photohadronic interactions also produce
cosmogenic neutrinos peaking around $10^9$ GeV. We test whether this model is
still viable in light of recent UHECR spectrum measurements from the Telescope
Array experiment, and upper limits on the cosmogenic neutrino flux from
IceCube. While two-parameter fits have been already presented, we perform a
full scan of the three main physical model parameters: source redshift
evolution, injected proton maximal energy, and spectral index. We find
qualitatively different conclusions 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. The predicted cosmogenic neutrino flux exceeds the
IceCube limit for any parameter combination. As a result, the proton dip model
is challenged at more than 95\% C.L. This is strong evidence against this model
independent of mass composition measurements.
