  Axion-photon conversion induced by intergalactic magnetic fields has been
proposed as an explanation for the dimming of distant supernovae of type Ia SNe
Ia) without cosmic acceleration. The effect depends on the intergalactic
electron density $n_e$ as well as the $B$-field strength and domain size. We
show that for $n_e\alt 10^{-9} {\rm cm}^{-3}$ the same mechanism would cause
excessive spectral distortion of the cosmic microwave background (CMB). This
small-$n_e$ parameter region had been left open by the most restrictive
previous constraints based on the dispersion of quasar (QSO) spectra. The
combination of CMB and QSO limits suggests that the photon-axion conversion
mechanism can only play a subleading role for SN Ia dimming. A combined
analysis of all the observables affected by the photon-axion oscillations would
be required to give a final verdict on the viability of this model.
