Then that is saying that the cultures which are most repressive towards women will win since those are the one's with the highest birth rates.
Maybe that is what will happen, at least in the medium term (100 years or so). History is the record of the expansion and contraction of different populations. The defeat of a state with a falling population by one with an increasing population is a major feature.
Keep an eye on what happens with China and Russia in the next 20 years.
On the other hand, the United States and Western/Central Europe once had healthy birthrates, without being significantly "repressive." There are many factors.