Population decline caused by low fertility has leads to skill loss and productivity decline consistently throughout history. From Tasmania's pre-history to the Greek Dark Ages to the fall of Rome and the European dark ages
I am going to need citations of studies that prove this statement.
In the mean time I have couple of objections:
The claim that fertility rates lead to decline in populations in the past is questionable at best. Only in modern times humanity has introduced effective birth control methods.
Population declines in the past are more likely linked to the destruction of individual civilizations due to war, famine caused by extended draught and epidemics or to the exhaustion in the available supply of slave labor (which was the bedrock of the economy of ancient mediterranean civilizations).
The article that you cited shows a lot of correlations but we all know that correlations do not equal causation.
Another objection that I have is that you equate productivity to skills of the workforce (which is part of it) but the higher economic outputs that we have seen in the last two centuries have more to do with innovations in technology.
You are right in asserting that I assume that productivity will continue to increase...the pace of technological improvements keeps accelerating on a global scale and shows no signs of stopping any time soon (barring a black swan event).
I'll do a full post about this with the sources and will link it here when ready.