  The extreme, time-variable Faraday rotation observed in the repeating fast
radio burst (FRB) 121102 and its associated persistent synchrotron source
demonstrates that some FRBs originate in dense, dynamic and possibly
relativistic magneto-ionic environments. Here we show that besides rotation of
the linear-polarisation vector (Faraday rotation), such media can generally
convert linear to circular polarisation (Faraday conversion). We use
non-detection of Faraday conversion, and the temporal variation in Faraday
rotation and dispersion in bursts from FRB\,121102 to constrain models where
the progenitor inflates a relativistic nebula (persistent source) confined by a
cold dense medium (e.g. supernova ejecta). We find that the persistent
synchrotron source, if composed of an electron-proton plasma, must be an
admixture of relativistic and non-relativistic (Lorentz factor $\gamma<5$)
electrons. Furthermore we independently constrain the magnetic field in the
cold confining medium, which provides the Faraday rotation, to be between $10$
and $30\,$mG. This value is close to the equipartition magnetic field of the
confined persistent source implying a self-consistent and over-constrained
model that can explain the observations.
