I seem to remember when that theory was discussed, it was explained that GPS clocks already take into account relativistic motion. Or something like that, don't quote me, I'm not a physicist.
[1] http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps....
"Particle Physics for Non-Physicists: A Tour of the Microcosmos"
http://www.thegreatcourses.com/tgc/courses/course_detail.asp...
But measuring the simultaneity of an event between two different points with a moving clock is a lot trick.
But they weren't. By my understanding, they were merely using the satellite signals to synchronize the ground clocks in the ground frame, and after that was done, the satellite's were never heard from again.