The origin and properties of the source of positrons annihilating in the Galactic Centre (GC) are still a mystery. One of the criteria that may discriminate between different mechanisms of positron production there is the injection energy of positrons. Beacom and Yuksel suggested a method to estimate this energy from the ratio of the 511-keV line to the MeV in-flight annihilation fluxes. From COMPTEL data, they derived that the maximum injection energy of positrons should be about several MeV. This significantly decreased the class of models of the positron origin in the GC, assuming that positrons lose their energy through Coulomb collisions only. However, observations show that the strength of the magnetic field in the GC is much higher than in other parts of the Galaxy; in the GC, it may range from 100 mu G to several mG. In these conditions, the synchrotron losses of positrons are significant and this extends the range of acceptable values of the injection energy of positrons. We show that if positron injection in the GC is non-stationary and the magnetic field is higher than 0.4 mG, both radio and gamma-ray restrictions permit the energy to be higher than several GeV.