This contradicts the accuracy stated at the beginning. Omega can't leave both boxes empty for people who try to adopt a mixed strategy AND also maintain his 99.whatever accuracy on one-boxers.

He can maintain his 99% accuracy on deterministic one-boxers, which is all that matters for the hypothetical.

Alternatively, if we want to explicitly include mixed strategies as an available option, the general answer is that Omega fills the box with probability = the probability that your mixed strategy one-boxes.