I have always wondered whether something like this could be observed.

I wonder, shouldn't this be already observable? I would look not at the median/mean fertility, but on the outliers on the upper end. If the share increases/becomes more expressed in social groups that are not associated with a large number of children otherwise, one should see indicators that the theory is has some thruth.

Also, we are talking about heritability and entire generations. If such trend can be identified, I would expect it to be very small and take time (measured in generations) until it will have a real effect.