  The dip model assumes that the ultra-high energy cosmic rays (UHECRs) above
10$^{18}$ eV consist exclusively of protons and is consistent with the spectrum
and composition measure by HiRes. Here we present the range of cosmogenic
neutrino fluxes in the dip-model which are compatible with a recent
determination of the extragalactic very high energy (VHE) gamma-ray diffuse
background derived from 2.5 years of Fermi/LAT data. We show that the largest
fluxes predicted in the dip model would be detectable by IceCube in about 10
years of observation and are within the reach of a few years of observation
with the ARA project. In the incomplete UHECR model in which protons are
assumed to dominate only above 10$^{19}$ eV, the cosmogenic neutrino fluxes
could be a factor of 2 or 3 larger. Any fraction of heavier nuclei in the UHECR
at these energies would reduce the maximum cosmogenic neutrino fluxes. We also
consider here special evolution models in which the UHECR sources are assumed
to have the same evolution of either the star formation rate (SFR), or the
gamma-ray burst (GRB) rate, or the active galactic nuclei (AGN) rate in the
Universe and found that the last two are disfavored (and in the dip model
rejected) by the new VHE gamma-ray background.
