Disentangling density, economic development, and education in terms of reduced fertility

From Population Density is a Key Factor in Declining Human Fertility by Wolfgang Lutz, Maria Rita Testa, and Dustin J. Penn. From the Abstract:

In studying the complex determinants of human fertility, social scientists have given little attention to population density, although reproduction has been shown to be density-dependent for a wide variety of other species. Using fixed effects models on the time series of 145 countries and controlling for key social and economic variables, we find a consistent and significant negative relationship between human fertility and population density. Moreover, we find that individual fertility preferences also decline with population density. These findings suggest that population density should be included as a variable in future studies of fertility determinants.

Urbanization increases density and cities are notorious consumers of people. In fact, I wonder if there are any major cities which have a positive native (i.e. non-immigrant and non-local migration from suburbs or country) population growth. None that I can think of.

We have been urbanizing rapidly and passed the 50% mark for urbanization about ten years ago, i.e. more than fifty percent of the worlds population now lives in cities.

Economists have long been aware of the association between economic development in general and universal education in particular as factors which drive down human fertility. Obviously economic development overlaps as a factor with densification. I wonder what the relative causal weight can be attributed to density over development and education. I am guessing it might be fairly significant.