All conscious beings should expect to experience the next moment in at least one Everett branch even if they stop existing in all other branches, and the moment after that in at least one other branch, and so on forever.

I don't think this is quite true. Rather, if there is precisely one Everett branch in which a conscious being still exists, they should expect to experience it. If there is no such Everett branch, the conscious being will still be conscious in this moment but will not experience any future moments. If there are multiple, the conscious being should expect to experience one of them.