There's basically two ways this plays out. First the Epstein class wins. If that happens we become a techno feudal hellscape and you'll have a about 50 to 60,000 people at the top and then everyone else will be living in abject poverty at the level of medieval times. Disease and starvation will wipe most of them out and keep the numbers down.
The other less likely possibility is that we go with a cooperative global civilization. You know that cookie spooky socialism crap. If that happens you will quickly see the population not just stop growing but shrink. People who aren't desperately trying to crank out babies without access to birth control so they can use them as cheap labor to replace machines that don't exist just don't have a lot of kids. When you take away the economic incentive it turns out the birth rates drop faster than even the projections your citing.
Either way the reason we have this huge population boom is because we were in a very bizarre and unique situation where we had just enough technology that a primitive culture that squeezes out babies left and right wasn't dropping dead constantly from war and disease and general lack of medical Care. That's not sustainable and it's not going to last. The Epstein class won't allow it. So we either get rid of the Epstein class which does not seem to be on the table or we let them turn us into something worse than serfs