I'm not saying it will not happen, there are certainly clues pointing out in that direction so no need for "except" :). On the other hand there are scenarios compatible with the data you are mentioning that do not lead to population stabilization.

For example. Let's say that while the majority of the population (Population A) is reducing its fertility rate a subpopulation (Population B) keeps it at a higher rate. Let's suppose that both populations maintain this behavior for several generations. In this case we would see an initial reduction in fertility rate not leading to a stabilization of the population

Will this happen?, I don't know. What I'm saying is that I feel in these discussions often people place too much faith on their preferred extrapolations ;)