Would Becker Barro and other economists agree that child support laws reduce fertility of rich men?

If so how.

Take into account

1. Money is paid to the mother and mostly can be used to make the mother happy instead of his son
2. Amount is proportional to income and something the man can't negotiate before conception
3. Money can be used to be spent on things father disagree, like gender reassignment
4. Provide very strong incentives for women to leave rich men and take his son. So that may discourage rich men from having more children.

Gemini says yes. But I want to know what economists think.

What I mean by reducing fertility, I do not mean necessarily reducing fertility to 0. A rich man that may want to have 20 children can end up having only 3 and go to jail or lost a lot of money because of that.