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It is the same wavefunction, and it is just being "sliced more thinly".
Do you think the number of worlds is a definite and objective fact, or that it depends on how you slice the wavefunction?
Well, it isn't really about what I think, but about what MWI is understood to say.
According to MWI, the worlds are being "sliced more thinly" in the sense that the total energy of each depends on its probability measure, and when a world splits its probability measure, and therefore energy, is shared out among the worlds into which it splits. The answer to your question is a "sort of yes" but I will qualify that shortly.
For practical purposes, it is a definite and objective fact. When two parts of the wavefunction have become decoherent...