(Abridged) Data collected by the Pierre Auger Observatory (Auger) provide evidence for anisotropy in the arrival directions of cosmic rays (CRs) with energies >57 EeV that suggests a correlation with the positions of AGN located within ~75 Mpc. A detailed study of the sample of AGN whose positions are located within 3.2 degrees of the CR events (and extending our analysis out to ~150 Mpc) shows that most of them are classified as Seyfert 2 and low-ionization nuclear emission-line region (LINER) galaxies whose properties do not differ substantially from other local AGN of the same types. Therefore, if the production of the highest energy CRs is persistent in nature, i.e., operates in a single object on long (>Myr) timescales, the claimed correlation between the CR events observed by Auger and local active galaxies should be considered as resulting from a chance coincidence. Additionally, most of the selected sources do not show significant jet activity, and hence, in most conservative scenarios, there are no reasons for expecting them to accelerate CRs up to the highest energies, ~10^20 eV. A future analysis has to take into account AGN morphology and may yield a correlation with a larger deflection angle and/or more distant sources. We further argue that the nearby radio galaxy NGC 5128 (Cen A) alone could be associated with at least 4 events due to its large radio extent, and PKS 1343-60 (Cen B), another nearby radio galaxy, can be associated with more than 1 event due to its proximity to the Galactic plane and, correspondingly, the stronger Galactic magnetic field the UHECRs encounter during propagation to the Earth. Future gamma-ray observations (by, e.g., Fermi Gamma-ray Space Telescope, and HESS) may provide additional clues to the nature of the accelerators of the UHECRs in the local Universe.
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