The excess of neutrino candidate events detected by IceCube from the direction of TXS 0506+056 has generated a great deal of interest in blazars as sources of high-energy neutrinos. In this study, we analyze the publicly available portion of the IceCube dataset, performing searches for neutrino point sources in spatial coincidence with the blazars and other active galactic nuclei contained in the Fermi 3LAC and the Roma BZCAT catalogs, as well as in spatial and temporal coincidence with flaring sources identified in the Fermi Collaborations All-Sky Variability Analysis (FAVA). We find no evidence that blazars generate a significant flux of high-energy neutrinos, and conclude that no more than 5-15% of the diffuse flux measured by IceCube can originate from this class of objects. While we cannot rule out the possibility that TXS 0506+056 has at times generated significant neutrino emission, we find that such behavior cannot be common among blazars, requiring TXS 0506+056 to be a rather extreme outlier and not representative of the overall blazar population. The bulk of the diffuse high-energy neutrino flux must instead be generated by a significantly larger population of less-luminous sources, such as non-blazar active galactic nuclei.
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