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.
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