  We report on our efforts to test the Einstein Equivalence Principle by
measuring the gravitational redshift with the VLBI spacecraft RadioAstron, in
an eccentric orbit around Earth with geocentric distances as small as $\sim$
7,000 km and up to 350,000 km. The spacecraft and its ground stations are each
equipped with stable hydrogen maser frequency standards, and measurements of
the redshifted downlink carrier frequencies were obtained at both 8.4 and 15
GHz between 2012 and 2017. Over the course of the $\sim$ 9 d orbit, the
gravitational redshift between the spacecraft and the ground stations varies
between $6.8 \times 10^{-10}$ and $0.6 \times 10^{-10}$. Since the clock offset
between the masers is difficult to estimate independently of the gravitational
redshift, only the variation of the gravitational redshift is considered for
this analysis. We obtain a preliminary estimate of the fractional deviation of
the gravitational redshift from prediction of $\epsilon = -0.016 \pm 0.003_{\rm
stat} \pm 0.030_{\rm syst}$ with the systematic uncertainty likely being
dominated by unmodelled effects including the error in accounting for the
non-relativistic Doppler shift. This result is consistent with zero within the
uncertainties. For the first time, the gravitational redshift has been probed
over such large distances in the vicinity of Earth. About three orders of
magnitude more accurate estimates may be possible with RadioAstron using
existing data from dedicated interleaved observations combining uplink and
downlink modes of operation.
