>You can try, but every study I've seen has found that men are favored in contested custody proceedings.
Do they? Every study I have seen has never accounted for the differences in the average case where the man challenges vs. the average case where they do not. Perhaps they are discriminated against to the point that they only seek to challenge the status quo when they have a very good case, which is why they win more often when they do challenge.
Its true that women are more likely to end up with custody, but that appears to because they are vastly more likely to seek custody.