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The incompatibility of general relativity and quantum theory is a fundamental problem in our understanding of the physical world. The Einstein equivalence principle (EEP) is a cornerstone of general relativity, and the gravitational redshift is a consequence of the local position invariance aspect of it. An accurate measurement of the gravitational redshift and a comparison with prediction is therefore of prime importance. Here we report on such a measurement. The Spektr-R spacecraft of the RadioAstron mission was launched in 2011 into a highly elliptical orbit around Earth. Since early on the spacecraft has been used for astrophysical observations of compact radio sources with the technique of space VLBI (very long baseline interferometry). The spacecraft has a hydrogen maser frequency standard on board that operated till 2017. For several years the downlink signals at 8.4 and 15 GHz, locked to the frequency of the hydrogen maser, were recorded at ground stations, which are also equipped with hydrogen masers. About 2900 sessions each with ∼100,000 frequency recordings at each of the two frequencies were obtained. During the 9-day elliptical orbits the spacecraft traveled through the varying gravitational potential of Earth, which according to the EEP should cause an oscillating gravi- tational redshift of the downlink signals. The predicted relative frequency shift varied between 6.8x10^{-10} and ~4x10^{-10}. The data from the RadioAstron mission are being largely independently analyzed by more than one group to provide for a better reliability of our anticipated final results. We report on our analysis of data recorded at the ground stations in Pushchino, Russia, and Green Bank, USA, and discuss the possibility of testing the EEP with a higher sensitivity than that obtained from the Gravity Probe A mission.