Limit law conjecture for reticulation nodes in d-combining tree-child networks

Let $d$-combining tree-child networks have $n$ leaves, and let $T_n^{(d)}$ denote the number of reticulation nodes in a network chosen uniformly at random from all such networks with $n$ leaves. A random variable $X_n$ converges weakly to a random variable $X$, written $X_n\stackrel{w}{\longrightarrow}X$, when its distribution converges weakly to the distribution of $X$.

Limit law conjecture. The following limit behavior holds:

$$n-1-T_n^{(2)}\stackrel{w}{\longrightarrow}\operatorname{Poisson}(1/2).$$

Here, $\operatorname{Poisson}(\alpha)$ denotes the Poisson distribution. For $d\geq 3$, the limit distribution of $n-1-T_n^{(d)}$ is degenerate.

This conjecture proposes limit laws for the number of reticulation nodes in general $d$-combining tree-child networks. It asserts a Poisson limit in the bicombining case and a degenerate limit for $d\geq 3$, complementing the paper's asymptotic counting results; the source does not provide evidence that the conjecture has been resolved.