[Quark2005]

There is a fairly respectable group of physics people who say that the observed behavior of GPS satellites violates Relativity theory.

I don't think that's true. GPS positioning uses a leading order correction based on General Relativity to compensate their positioning. Without it the satellites would lose several meters of accuracy every day.

[Iris7]

I am not in anything like a position to say yea or nay here, for sure.

My impression is that the correction you refer to is empirical in major part and is something of an exercise in curve fitting.

Not claiming special insight here. I just suspect that True Knowledge (term borrowed from Iain Banks, suggests human frailty and the need for enough certainty to have at least some hope for the future, and is a mixture of serious and ironical) is the pot of gold at the end of the rainbow where we endlessly pursue and endlessly do not find. A "chimera", though the analogy is a bit strained.