In this work, we present an estimate of the cosmic-ray mass composition from the distributions of the depth of the shower maximum ($X_\mathrm{max}$) measured by the fluorescence detector of the Pierre Auger Observatory. We discuss the sensitivity of the mass composition measurements to the uncertainties in the properties of the hadronic interactions, particularly in the predictions of the particle interaction cross-sections. For this purpose, we adjust the fractions of cosmic-ray mass groups to fit the data with $X_\mathrm{max}$ distributions from air shower simulations. We modify the proton-proton cross-sections at ultra-high energies, and the corresponding air shower simulations with rescaled nucleus-air cross-sections are obtained via Glauber theory. We compare the energy-dependent composition of ultra-high-energy cosmic rays obtained for the different extrapolations of the proton-proton cross-sections from low-energy accelerator data.