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Absolutely possible, might even be a good idea, but my expectation is that the results won't be robust: the fakes will be uncovered by a slightly differently trained classifier. Maybe even the same classifier with a different random initialization.

Sounds like overfitting the defense against classification. Would existing solutions to overfitting possibly fix this (though make such a network even more expensive to train)?

Maybe avoiding detection would make the generated images look more realistic as a side effect.

Sounds possible