It holds. It holds practically everywhere on Earth. Income is one of the strongest inverse correlations to fertility.
https://bmcpublichealth.biomedcentral.com/articles/10.1186/s...
https://bmcpublichealth.biomedcentral.com/articles/10.1186/s...
https://ifstudies.org/blog/more-money-more-babies-whats-the-...
https://www.demographic-research.org/articles/volume/51/26
https://www.sas.upenn.edu/~jesusfv/Slides_London.pdf
(your paper is five years old, and is lagging broad, rapid global fertility decline trends; it's not income-fertility, its "educated, empowered women with access to birth control have less kids, no kids, and/or delay childbirth" regardless of income, with the caveat being some higher income cohorts and ultra orthodox religions/cultures [Israel] having higher than baseline fertility)
Education and contraceptives as the article I linked are separately correlated and powerful, as is religious adherence. They're all factors. Note that each of these factors is differently important in different countries or regions, so just because you found a study that shows a positive correlation in the Netherlands that it is in isolation important or that it would work the same way in the US.