  Using the Auger mass-composition analysis of ultra high energy cosmic rays,
based on the shape-fitting of $X_{max}$ distributions, we demonstrate that mass
composition and energy spectra measured by Auger, Telescope Array and HiRes can
be brought into good agreement. The shape-fitting analysis of $X_{max}$
distributions shows that the measured sum of proton and Helium fractions, for
some hadronic-interaction models, can saturate the total flux. Such p+He model,
with small admixture of other light nuclei, naturally follows from cosmology
with recombination and reheating phases. The most radical assumption of the
presented model is the assumed unreliability of the experimental separation of
Helium and protons, which allows to consider He/p ratio as a free parameter.
The results presented here show that the models with dominant p+He composition
explain well the energy spectrum of the dip in the latest (2015 - 2017) data of
Auger and Telescope Array, but have some tension at the highest energies with
the expected Greisen-Zatsepin-Kuzmin cutoff. The Auger-Prime upgrade experiment
has a great potential to reject or confirm this model.
