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We report on preliminary results of the gravitational redshift test performed with the space radio telescope RadioAstron, which has been in a highly eccentric orbit around the Earth since 2011. We probed the flow of time on board, which varies while RadioAstron passes through the varying gravitational potential of the Earth, by recording the spacecraft’s downlink signal synchronized to its on-board H-maser. A total of 17 successful experiments were performed, each comprised of several observations at various distances between the spacecraft and the Earth. The experiment requires us to extract the tiny gravitational redshift signal from the mixture of much larger effects, including the nonrelativistic Doppler, ionospheric and tropospheric frequency shifts, higher-order kinematic effects of special relativity, Earth tides, and various instrumental effects. Preliminary analysis of the data of two experiments gives us a fractional accuracy of the gravitational redshift test of order 10^{−4}. We present arguments in favor of the experiment accuracy currently being dominated by that of the ionospheric frequency shift and discuss prospects for improving this result further.