The answer that seems to be emerging from several different lines of research is that a) they always had fairly low fertility and b) they didn't really go extinct as such, they just intermixed with Homo Sapiens Sapiens and because the later had much higher fertility, Neanderthal genes got diluted down to the present ~2% in the Eurasian population.
Sounds plausible indeed. Anyways, neanderthals operating a large scale fat production 125 thousand years ago could be a good plot for another hollywood movie scenario. Any takers?
You might enjoy Hominids by Robert Sawyer
Tangentially related, The Man from Earth is really good as well.
Very few films choose to shoot on a camcorder, and fewer still pull it off well.
I thought even after the merge the Neanderthal genes continued to get rarer, indicating natural selection against them
If it's 2% now after 2000-3000 generations, it must have stabilized because any number <1 is basically zero when raised to the 2000th power.