It's all theoretical of course, so none of this has been proven through experiments or people actually doing this.

well, proven every time you successfully use a GPS for navigation

there are two effects, the atomic clock in the GPS satellite ticks slower because its moving fast in the orbit, but also faster because the clock is farer out of the gravity field of the planet then the clock on the surface

in summation this has to be taken into account or would lead to an offset in the position calculation of up to 11km per day , and adding up each day