There is convincing quantitative evidence that all of the extra-galactic radio sources radiate by the commonly accepted incoherent synchrotron process, or a mathematically similar mechanism such as synchro-Compton radiation. This includes:
The shapes of the spectra of the extended (transparent) sources are power law or dual power, and their detailed shapes are in agreement with synchrotron models where the relativistic particles both gain and lose energy.
In the compact sources the spectral peak occurs at shorter wavelengths in the smaller sources, as predicted by the synchrotron model, and the measured angular sizes are in good agreement with those estimated from the observed self-absorption cut-off wavelength.
The maximum observed brightness temperature is ~ 1012 °K, as is expected from an incoherent synchrotron source, which is "cooled" by inverse Compton scattering.
The variations in intensity and polarization and their dependence on wavelength and time are in good agreement with those expected from an expanding cloud of relativistic particles.
The outstanding problems in the understanding of the radio galaxies and quasars are:
the source of energy and its conversion to relativistic particles
the connection, if any, between the compact opaque sources and the extended transparent one
the evolution and confinement of the extended sources.
The third and possibly the second problem present challenging complex problems in magneto-hydrodynamics and plasma physics, while the first may well require fundamentally new physics before it can be understood.