The average shower depth $\langle X_\text{max}\rangle$ of Ultra-high Energy Cosmic Rays is observed to flatten with energy at the highest energies. The standard interpretation of these data is that the composition is getting heavier; an alternative interpretation is the existence of new effects in proton interactions above $\sim 50$ TeV center-of-mass energy. We have used CORSIKA to study, through air-shower simulations, observational signatures of a possible increase in cross-section and multiplicity in collisions exceeding this threshold. We have simulated hadronic collisions for primaries with energies in the range $10^8 - 10^{11}$ GeV. We have used two different high energy models for the simulations, QGSJETII-04 and EPOS LHC, with Fluka for low energy interactions on both. A smooth transition from Galactic to extragalactic cosmic rays was implemented, by fitting a Galactic component with an exponential suppression at $\sim 10^9$ GeV. The remaining flux in Auger data was interpreted as extragalactic protons. Above $10^{9}$ GeV, the proton-air cross-section and the multiplicity of secondary particles were altered, so as to bring the simulated $\langle X_\text{max}\rangle$ in agreement with Auger data. The parameter space of the viable cross-section and multiplicity in the scenario where the composition of Auger cosmic rays at the highest energies remains unchanged and light, places constraints on the phenomenology of any new physics affecting the interactions for high energy protons that may be probed by $\sqrt{s}>50$ TeV collisions. We found out that if new physics indeed sets in, the cross-section of proton-Air interactions has to be $\sim 800-900$ mb at $140$ TeV center-of-mass energy, accompanied with an increase of the number of secondary particles by a factor between $2-3$.