I started reading this, but stopped when it went into the It-can-be-read-anyway-we-want-to kinda stuff...
This was the most interesting read in a while.
This was, primarily, the most "compelling" evidence...
"The constellation of satellites orbits the earth at an altitude of about 20,000km. At this height, the atomic clocks on board the satellites run faster than the same clocks on earth by 42 microseconds per day. The satellites are travelling at around 12,000 km per hour which produces an opposing relativistic effect of slowing the clocks down by about 5 microseconds per day."
Atomic clocks work by measuring the microwave signal that electrons in atoms emit (9,192,631,770 Hz) when they change energy levels in caesium-133 or rubidium-87.
I really don't see that relatively low speed (compared to light) affecting it too much, especially when it is in a (loosely) fixed gravitational pull, but it could be gladly thrown in the mix.
Because of the gravitational mass difference (20,000 km), the clock that is closer to the gravitational mass (earth), i.e. deeper in its "gravity well", appears to go slower than the clock that is more distant (GPS).
So, while the gravitational pull affects mass of the caesium-133, allowing for more jumps in the electrons energy levels, all it does it measure more jumps.
Time didn't go "faster", the atomic clock measured more jumps, yielding in more units to count.
And more "time".