You can check that in the data easily. The moment a society becomes wealthy, the reproduction rate goes down. That's an undisputed fact in sociology.
I'm not denying that. But there are several complications.
First, I am calling into question how the context within which that "moment" occurs might matter. So if you've gone through a process of "development" for several decades that results in your country becoming "wealthy," is the result different than say if there was a large scale global redistribution of wealth that resulted in your society becoming wealthy "overnight."
Of course it's worth trying, but I am skeptical. I think that the effects of class linger beyond a change in wealth.
Second, a decline in birth rate is not enough. You need to reach 2 children or less worldwide to slow/reverse growth. So just to say that the data shows a decline does not guarantee it will be enough of a decline.
Third, there are many wealthy families that continue to have lots of children for religious reasons, for example. ***
In other words, if you've only got a very small window to turn things around, there are some lags here that might prove to be problematic.
The attached graph is not entirely relevant (since the countries aren't wealthy), but something can be gleaned from it, because it also includes data about the wealthy within those countries. So you can see that in all but two of these countries those women with higher education (which, as the percentages of the population indicate, are mostly wealthy) are still having more than 2 children, and half of those are having 3 or more.
*** addendum: as the graph shows, the countries with the highest birth rates (over 3.5) for women with higher education are Niger, Burundi, Afghanistan and Nigeria. Niger and Afghanistan are nearly 100% Muslim nations. Burundi is a large majority Catholic, and Nigeria is about half Muslim, and a good portion Catholic. Of course other cultural factors are relevant.