That statement is not only true, but a tautology. The multiplication will only cease if the fertility rate declines. For traits and values that are highly heritable, these too will consequently increase in a rate that is, at the minimum, proportional to the rate of fertility. Especially in modern times this is literally how Islam is 'spreading', and it's spreading rapidly, everywhere.

And bear in mind that fertility is an exponential system, in both increases and decreases. So groups that are removing themselves from the gene pool will do so with a rapidity that is quite counter-intuitive. Like a fertility rate of 1 doesn't sound that insane (it obviously translates to literally every single woman having one child on average), yet it results in a generational decline of 50%, with a generation tending to be around our window of fertility - about 20 years.

So imagine two groups start at 16 people and one group maintains a fertility rate of 1, and the other group maintains a fertility rate of 3. After just 5 generations, about a century, the low fertility group will have 1 person, and the higher fertility group will have 81. You went from equal size to an 8100% difference, after a single century! And that's with fertility levels that are entirely realistic and not just comparing extremes like Nigeria or South Korea.

[1] - https://www.pewresearch.org/short-reads/2017/04/06/why-musli...