  The Einstein Equivalence Principle (EEP) is a cornerstone of general
relativity and predicts the existence of gravitational redshift. We report on
new results of measuring this shift with RadioAstron (RA), a space VLBI
spacecraft launched into an evolving high eccentricity orbit around Earth with
geocentric distances reaching 353,000 km. The spacecraft and ground tracking
stations at Pushchino, Russia, and Green Bank, USA, were each equipped with a
hydrogen maser frequency standard allowing a possible violation of the
predicted gravitational redshift, in the form of a violation parameter
$\varepsilon$, to be measured. By alternating between RadioAstron's frequency
referencing modes during dedicated sessions between 2015 and 2017, the recorded
downlink frequencies can essentially be corrected for the non-relativistic
Doppler shift. We report on an analysis using the Doppler-tracking frequency
measurements made during these sessions and find $\varepsilon = (2.1 \pm
3.3)\times10^{-4}$. We also discuss prospects for measuring $\varepsilon$ with
a significantly smaller uncertainty using instead the time-domain recordings of
the spacecraft signals and envision how $10^{-7}$ might be possible for a
future space VLBI mission.
