The measurements by the Pierre Auger Observatory of the energy spectrum and mass composition of cosmic rays can be interpreted assuming the presence of two extragalactic source populations, one dominating the flux at energies above a few EeV and the other below. To fit the data ignoring magnetic field effects, the high-energy population needs to accelerate a mixture of nuclei with very hard spectra, at odds with the approximate $E^{-2}$ shape expected from diffusive shock acceleration. The presence of turbulent extragalactic magnetic fields in the region between the closest sources and the Earth can significantly modify the observed CR spectrum with respect to that emitted by the sources, reducing the flux of low-rigidity particles that reach the Earth. We here take into account this magnetic horizon effect in the combined fit of the spectrum and shower depth distributions, exploring the possibility that a spectrum for the high-energy population sources with a shape closer to $E^{-2}$ be able to explain the observations. We find that a large inter-source separation $d_{\rm s}$ and a large magnetic field RMS amplitude within the Local Supercluster region, such that $B_{\rm rms}\simeq 100\,{\rm nG}\,(40\,{\rm Mpc}/d_{\rm s})\sqrt{25\,{\rm kpc}/L_{\rm coh}}$, are needed to interpret the data within this scenario, where $L_{\rm coh}$ is the magnetic field coherence length.