How accurately has general relativity's prediction of gravitational redshift been measured?

A recent test (2018) using atomic clocks aboard two satellites found general relativity's gravitational redshift prediction to be accurate to $(+0.19 pm 2.48)times10^{-5}$.

This was not a planned experiment and the satellites had accidentally been delivered on elliptic rather than circular orbits in 2014. They were useless for their original purpose of navigation. Instead of taking a complete loss, the satellites were repurposed for experimentation.

Previously the most accurate measurement was taken in 1976 by Gravity Probe A which had an accuracy of 70 parts per million. The new test improves the accuracy by a factor of 5.6.

See the paper, "A gravitational redshift test using eccentric Galileo satellites".