We have a bonus question winner ... and yes you got all the various components and pass with an A+.
I did limit the question specifically to GR for a reason
So the answer is sort of conditionally yes under GR although as you said you can sort of argue condition exemptions etc. One of the problems is under GR there are many possible distinct definitions of the gravitational stress–energy–momentum pseudotensor because it is a very mathematical process which you also got. So the GR answer you derive may actually hinge on what definition you selected and it's funny watching two scientists go at it with different definitions
You got the final part that the actual true answer probably needs a full theory of gravity, so at the moment all we can say is probably yes without the full theory or an experimental result.
So you got every component that the question poses and did better than some scientists I have seen tackle the question. It is a really challenging question that usually sets of furious debate and is really fun to ask a particular group of physicists
Now I should say my GR is not up to date and it really isn't my thing and this may be a really interesting question to put to Ethan Siegel to get a really modern view on this.
