Crump-Mode-Jagers branching processes

Much of my research concerns analysis of Crump-Mode-Jagers branching processes, which model the growth of a population over time. Individuals possess `birth times' indicating when they were born. If the average number of offspring per child exceeds one, it is well-known that the population grows indefinitely with positive probability. This is pictured on the left. If it is at most one, the population eventually dies out. This is pictured on the right. 

In the trees describing the evolution of these processes, under  what circumstances is there a single node which possesses the maximal number of connections over time? Sufficient criteria for the emergence of such a node (a persistent hub), or the non-emergence of such a node, have been proved here: arXiv.2410.24170. These two phases are illustrated below. On the left, the process is guaranteed to eventually possess a persistent hub. On the right, it is guaranteed that infinitely many new-nodes `take-over' the maximal degree.