For understanding the diversity of jetted active galactic nuclei (AGN) and especially the puzzling wide range in their radio-loudness, it is important to understand what role the magnetic fields play in setting the power of relativistic jets in AGN. We have performed multi-frequency (4-24 GHz) VLBA phase-referencing observations of the radio-intermediate quasar III Zw 2 using three nearby calibrators as reference sources to estimate jet magnetic flux by measuring the core-shift effect. By combining the self-referencing core-shift of each calibrator with the phase-referencing core-shifts, we obtained an upper limit of 0.16 mas for the core-shift between 4 and 24 GHz in III Zw 2. By assuming equipartition between magnetic and particle energy densities and adopting the flux-freezing approximation, we further estimated the upper limit for both magnetic field strength and poloidal magnetic flux threading the black hole. We find that the upper limit to the measured magnetic flux is smaller by at least a factor of five compared to the value predicted by the magnetically arrested disk (MAD) model. An alternative way to derive the jet magnetic field strength from the turnover of the synchrotron spectrum leads to an even smaller upper limit. Hence, the central engine of III Zw 2 has not reached the MAD state, which could explain why it has failed to develop a powerful jet, even though the source harbours a fast-spinning black hole. However, it generates an intermittent jet, which is possibly triggered by small scale magnetic field fluctuations as predicted by the magnetic flux paradigm of Sikora & Begelman (2013). We propose here that combining black hole spin measurements with magnetic field measurements from the VLBI core-shift observations of AGN over a range of jet powers could provide a strong test for the dominant factor setting the jet power relative to the accretion power available.
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