Can a gamma-ray dim radio blazar produce a 200-PeV neutrino? The case of PMN J0606-0724 and KM3-230213A

An extremely energetic muon has been recently detected by the Cubic Kilometre Neutrino Telescope (KM3NeT), indicating the observation of a neutrino with the estimated energy of $\left( 2.2^{+5.7}_{-1.0} \right)\times 10^{17}$~eV. Radio blazar PMN~J0606$-$0724, not detected in gamma rays, is located within the reported error region of the neutrino arrival direction, and was flaring at the time of the event. Here we demonstrate that the neutrino could be produced in a photohadronic interaction in its radio core. The necessary proton power is of order of the source's photon luminosity, and protons can be accelerated to the required energies in the core, while high-energy gamma rays cannot leave the source because of intense production of electron-positron pairs. Expected contribution of the population of similar flaring sources matches non-observation of energetic events by other neutrino telescopes.