This is a nice writeup, but I feel it gives too strong an impression that Einstein's General Relativity completely incorporates "Mach's principle" with the solutions of Brill, Cohen and onwards with frame dragging.
To be clear, frame dragging is clearly a 'Machian' effect, but the magnitude is generally very small as a rotating body only drags the frames 'a little'. The only truly Machian version is for the 'extremal' case in which the frame dragging is complete.
In essence: The fact that GR admits solutions like Brill-Cohen which are asymtotically Minkowski is in contradiction with a full interpretation of Mach's principle. There is a lot of subtlety in defining what is meant by all of this, though, even among those practitioners who strive to further embody something called 'Mach's Principle', see: https://www.amazon.com/Machs-Principle-Newtons-Quantum-Einstein/dp/0817638237
Oddly enough it's not really clear. Einstein would have certainly hoped not. He believed all motion was relative and tried to incorporate Mach's principle - that inertial frames (and therefore the existence of "absolute rotation") is determined by the distribution of matter - into general relativity. Whether or not he succeeded is an open question. This book is a massive collection of work on this topic, and the conclusion is essentially "I dunno". It depends on precisely how you define the question and on the exact nature of our universe, and even then there are disagreements on the answer