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I disagree with Carl's description of the work.
The question of which of these is
relevant can be settled by experiment, but the experiment
has not been done and is not easily doable.
But let me quickly point out that [1] has problems of its
own. It protects Einstein's principle of equivalence, at
the cost of violating Galileo's principle of relativity!
It is amusing that there could exist such a simple physics
question for which no cut-and-dried answer is known.
But I suspect whenever you have two formulas that differ by an
integration by parts, you can create some kind of mystery
by pushing the boundary terms into the far past and future.