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What cosmological assumptions? Assumptions related to identity, perhaps, as discussed here. But it seems to me that MWI essentially guarantees that for every observer-moment, there will always exist a "subsequent" one, and the same seems to apply to all levels of a Tegmark multiverse.
I don't think MWI is sufficiently well defined or understood for it to be known whether or not that is implied. For example it would not be the case in Robin Hanson's mangled worlds proposal, and no one knows whether that proposal is correct or not.