All reputable biologists and sociologists would point out that extrapolating population trends for cultural memes such as “liberalism” and “religion” is nothing like extrapolating future population growth based on purely genetic traits such as eye colour. There are so many variables and unforeseen circumstances that affect the growth or decline of any particular cultural meme that extreme caution should always apply to calculations of population trends for these memes beyond the very near future. Those that refuse to be cautious usually end up with a lot of egg on their faces, as history has proven again and again. One important example that undermines the entire premise (that increased birth rates = increased cultural dominance) is the example of Catholicism in England.
There was a lot of fear and prejudice directed towards Catholics living in England after the establishment of the Anglican Church in the 16th Century. This was based on a simple extrapolation that since Catholics easily outnumbered Protestants per family it was inevitable that Catholics would replace Protestants in the population until they reached a tipping point a few generations hence and could take the country back for the Pope. This however is not what happened, and shows the fallacy of treating memes the same as genes. Not every Catholic child born into a Catholic family remained Catholic. Many converted to the Church of England, either through conviction, a desire to get ahead in an Anglican state, or through inter-marriage and so the percentage of Catholics in the population increased a lot slower than was predicted and then unforeseen circumstances occurred that affected all religions in the UK, which were the historical developments of the Enlightenment, the Industrial revolution and then the rise of secularism and multiculturalism.
Catholics in Britain still have larger families than most groupings in the UK, but they are in no more danger of taking over the population through birth-rate than they ever were - not that nowadays they are minded to seek power as Catholics first. This example has been repeated again and again throughout history regardless of whether the meme with the higher birth rate was Catholics, Orthodox Jews, or Conservatives.
If you are talking about genes, rather than memes, then the negativists’ claims become more possible but even then, as the history of America has shown, ethnic and immigrant inter-marriage tends to be a natural deflator of even these population trends as the effect of the melting pot takes hold.