The paper you link makes the case that this hypothesis has no real evidence to stand on. I'm not sure how concerned anyone should be about this particular brand of fortune-telling.

Wikipedia states this about the referenced 'demographic transition' model. Note the last sentence:

> It must be remembered that the DTM is only a model and cannot necessarily predict the future. It does however give an indication of what the future birth and death rates may be for an underdeveloped country, together with the total population size. Most particularly, of course, the DTM makes no comment on change in population due to migration. It is not applicable for high levels of development, as it has been shown that after a HDI of 0.9 the fertility increases again.

You are right. I should have pointed this out. I was trying to find citation about the conjecture I mentioned. I remember reading somewhere but only found this which says that it is an assumption.