We explore the correlation of $gamma$-ray emitting blazars with IceCube neutrinos by using three very recently completed, and independently built, catalogues and the latest neutrino lists. We introduce a new observable, namely the number of neutrino events with at least one $gamma$-ray counterpart, $N_{ u}$. In all three catalogues we consistently observe a positive fluctuation of $N_{ u}$ with respect to the mean random expectation at a significance level of $0.4 - 1.3$ per cent. This applies only to extreme blazars, namely strong, very high energy $gamma$-ray sources of the high energy peaked type, and implies a model-independent fraction of the current IceCube signal $sim 10 - 20$ per cent. An investigation of the hybrid photon -- neutrino spectral energy distributions of the most likely candidates reveals a set of $approx 5$ such sources, which could be linked to the corresponding IceCube neutrinos. Other types of blazars, when testable, give null correlation results. Although we could not perform a similar correlation study for Galactic sources, we have also identified two (further) strong Galactic $gamma$-ray sources as most probable counterparts of IceCube neutrinos through their hybrid spectral energy distributions. We have reasons to believe that our blazar results are not constrained by the $gamma$-ray samples but by the neutrino statistics, which means that the detection of more astrophysical neutrinos could turn this first hint into a discovery.
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