Do multifrequency polarimetric observations of BL Lac rule out a hadronic origin for its X-ray emission?

Recent multifrequency polarimetric observations of the eponymous blazar BL Lac reveal an extremely large degree of polarization in the optical band (average of $25\%$, reaching $45\%$), together with a small ($\lesssim 7\%$) degree of polarization in the X-ray band. This has been interpreted as evidence that the X-rays are produced through inverse Compton emission by relativistic electrons, thus ruling out alternative models based on hadronic processes. Here we revisit the observational evidence, interpreting it in a framework where the observed radiation is entirely produced through synchrotron emission. Electrons produce the radio-to-optical component and protons produce the X-rays and the gamma-rays. We determine the jet magnetic fields from an MHD model of magnetically dominated stationary axisymmetric outflows, and show that the X-ray emission from the protons is naturally less polarized than the optical emission from the electrons. The model parameters required to reproduce the multifrequency polarimetric observations are fully compatible with blazar jets.