Solomonoff induction does need to be modified here to support multiple observers, but the modification is minor; we just need to allow a universe to specify a probability distribution over observers (this is basically the SSA; different modifications are needed for different anthropic theories). The results in the probability of each individual bitstring including a factor of 2^-1000000, but the summed probability - the probability of "QM is (approximately) true" - is much greater than the probability of any alternate explanation. Most of that probability mass comes from algorithmically random strings, so that is what you should expect.