Can you expand on the family court claims? From what I've heard, fathers who ask for custody are slightly favored over mothers in family court. Women only end up with custody so much more often because very few fathers ask for custody (and custody is usually not given to the unwilling).
Unfortunately most of the papers I can find are locked behind paywall (ex. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.174-1617.1...)