That's a good question, but I don't have a detailed answer. I'm almost certain that there will have been empirical studies of the kind you described, but since I'm not a specialist, I can't cite any specific ones.

I can speculate that, since most of us know how hard it can be to overcome our harmful habits, despite our cognitive awareness of them, it might also be hard to control effects of our cognitive biases even after becoming aware of them. But that's only a guess.

It's certainly a subject worth exploring further. Thanks. Maybe some other reader can provide a more specific answer.