Recursive limit-law conjecture for fringe patterns in ranked tree-child networks

A fringe pattern is a connected substructure of a ranked tree-child network that has entirely evolved from a fixed set of lineages by consecutively adding branching and reticulation events. Let $F$ be a fringe pattern. Denote by $P$, respectively $P_1$ and $P_2$, the patterns obtained from it by removing the last event; the second case occurs only if the last event is a reticulation event and removing it disconnects the pattern.

Recursive limit-law conjecture. The limit law of $F$ is determined recursively as follows. If $P$ is a normal pattern, then $F$ is a Poisson pattern; in all other cases for $P$, $F$ is a degenerate pattern. If $P_1$ and $P_2$ are both normal patterns, then $F$ is also a normal pattern; if one of $P_1$ and $P_2$ is normal and the other is Poisson, then $F$ is a Poisson pattern; in all remaining cases, $F$ is a degenerate pattern.

The conjecture extends the paper's classification of limit laws for patterns of heights one and two. It predicts that every fringe pattern has one of three asymptotic behaviors—normal, Poisson, or degenerate—determined by recursively analyzing the event that created it.