Wouldn't population growth rate need be massive in order to overcome the large difference in volume that a small increase to the diameter of the sphere causes?

Wouldn't population growth rate need be massive in order to overcome the large difference in volume that a small increase to the diameter of the sphere causes? Given a quick scan, I don't see anywhere that takes into account population increases in the paper. The only way I think this works is if you count meeting the society's Von Neumann probes as meeting them.