We study the propagation of ultrahigh-energy cosmic rays in the magnetized cosmic web. We focus on the particular case of highly magnetized voids (B ̃nG ), using the upper bounds from the Planck satellite. The cosmic web was obtained from purely magnetohydrodynamical cosmological simulations of structure formation considering different power spectra for the seed magnetic field in order to account for theoretical uncertainties. We investigate the impact of these uncertainties on the propagation of cosmic rays, showing that they can affect the measured spectrum and composition by up to ≃80 % and ≃5 %, respectively. In our scenarios, even if magnetic fields in voids are strong, deflections of 50 EeV protons from sources closer than ̃50 Mpc are less than 15° in approximately 10-50% of the sky, depending on the distribution of sources and magnetic power spectrum. Therefore, UHECR astronomy might be possible in a significant portion of the sky depending on the primordial magnetic power spectrum, provided that protons constitute a sizeable fraction of the observed UHECR flux.