I'm not going to find sources right now, but from my understanding the research shows that the greatest impact on number of children is education of girls. Once women have more options, staying home their whole life popping out babies seems less desirable.

There will no doubt be a push by some of the most conservative idiots to stop educating girls.

I'd argue that the minimum education level rising in general is already strongly correlated itself, because it indicates that "uneducated" children are economically worthless (=> parents need to pay more to educate and children take longer until self-sustainable and economic "worth" of adolescents is relatively lower).