People may indeed limit themselves to having only two, one, or even no children at all, but that will only lead to extinction of such self-limiting groups.

This is not "if" this is "when".

But it will take centuries, a lot things can change in the meantime, so we don't know what will happen in the end, but assuming constant growth when the growth in devleoped countries is over already is weird.

Prognoses for world population already show the growth stopping in next few decades.