Predictions this far out are pretty silly given how much technological and change we've had in the past century and how much we can see on the horizon. Still, even if you held our technological state constant a permanent gradual decline is very unlikely given biological and cultural evolution: the children of people who have more kids than average will likely do the same, undoing the demographic transition.
The article is only considering the social majority and ignoring the existing of minority groups – for example, ultra-Orthodox Jews – who appear to be consistently maintaining high birth rates and avoiding any demographic transition. While they are only a tiny minority in most countries (Israel being the exception), if they continue on their current trajectory for a few centuries, they (and similar groups) will come to make up the majority of the population.
Demographics are actually pretty stable and accurate long term predictors of lots of other things.
Now, birthrates might be something harder to predict, although they too are subject to demographic forces. (Less people have fewer babies even with higher fertility.) The problem will be slow to correct.
