Yes, because it wouldn't be possible to keep the environment "identical" for very long, so they would diverge. Thinking about it more... I don't even think it

Yes, because it wouldn't be possible to keep the environment "identical" for very long, so they would diverge. Thinking about it more... I don't even think it is remotely possible to control the "initial conditions" perfectly enough (upon waking up), so they would diverge instantly. Sure, they might act very similarly, but it would quite clear that they are separate entities.