I will give historical information
This historical information will pretty much have the effect of unquestioning the question.
What we know today under the name 'Mach's principle' was not proposed by Ernst Mach. It was first proposed as a principle by Einstein. From the start Einstein referred to the notion as 'Mach's Principle'. For context: historians of science describe that Einstein wasn't the only one at the time to do so, and that Einstein wasn't the first to refer to the notion as 'Mach's principle.
whatever the case, it is clear that Einstein referring to the notion as 'Mach's principle' made other physicist think that Ernst Mach had proposed it. But that is not the case.
Ernst Mach adhered to a very austere philosophy of physics. Mach advocated that the bar for accepting something as scientific fact or scientific principle should be very high. For instance, Ernst Mach was against regarding the existence of atoms as certain. Mach was aware of the circumstantial evidence, of course, but he argued that on the existing evidence the existence of atoms should not be declared scientifically certain.
In his book about newtonian mechanics Ernst Mach had discussed the observation that the inertial coordinate system of the solar system does not rotate with respect to the fixed stars. Mach declared that it was out of bound for science to speculate why that coincidence is there. Science should use it, of course, but should not speculate about the origin of it.
In the following I will refer to the notion as: Einstein's Mach's principle
Einstein used an interpretation of Einstein's Mach's principle along the following lines: the inertia of a Universe is generated by the combination of all matter in the Universe. That is, Einstein preferred to regard inertia as a particular instance of gravitational effect.
Einstein expected that GR embodies that interpretation of Einstein's Mach's principle as follows: GR applied to cosmology will not allow solutions that describe a Universe devoid of matter. Einstein expected: in order for a Universe to exist at all there must be matter in it.
Historicans of physics describe: a few years after the publication of GR the astronomer Willem de Sitter found a cosmological solution that describes a Universe with no matter in it.
Historians of physics describe that for a prolonged time Einstein made great efforts to show that de Sitter's solution was invalid.
It took a while, but in the end Einstein acknowledged: the solution that de Sitter had found was a valid solution.
Historians of physics describe that in the following years ceased to mention Einstein's Mach's principle altogether. By contrast, prior to the de Sitter solution Einstein had repeatedly asserted that Einstein's Mach's principle was a foundation of GR. Einstein never published an explicit retraction of Einstein's Mach's principle, the evidence is that he ceased mentioning it.
Many years later (1954) the physicist Felix Pirani, when interviewing Einstein, asked about Mach's principle, to which Einstein replied: "Really, we should not be talking about Mach's principle anymore."
von dem Machschen Prinzip sollte man eigentlich überhaupt nicht mehr sprechen.
(Quoted in Mathematische Probleme in der Einstein – de Sitter Kontroverse, author: Stefan Röhle. Röhle gives the Einstein biography by Pais as source.)
The fact that Einstein soon abandoned Einstein's Mach's principle does not necessarily mean that other physicists should abandon the notion too, but still: it is relevant.
There are multiple interpretations/formulations of Einstein's Mach's principle in circulation. The question is not so much whether GR embodies Einstein's Mach's principle. Rather the question is: which version/interpretation of Einstein's Mach's principle, if any, is GR compliant with?
