Critical phase-transition conjecture for parking on a random tree

Let $N$ be the offspring variable with mean $\lambda=1$, variance ${\operatorname{var}(N)}\leq 1$, and let $P$ be a nonnegative integer-valued parking variable whose distribution is stochastically increasing in its mean $\alpha={\mathbb E}[P]$. Assume ${\operatorname{var}(P)}<\infty$ for every $\alpha\geq 0$, and define

$$h(\alpha)={\operatorname{var}(P)}+\alpha^2-\alpha={\mathbb E}[P^2]-\alpha.$$

Define

$$\alpha_c=\inf\left\{\alpha\geq 0:\alpha=1-\sqrt{{\operatorname{var}(N)}h(\alpha)}\right\}.$$

Critical phase-transition conjecture. The expected number of cars that can leave the root, $X$, satisfies

$${\mathbb E}[X]=\begin{cases}\displaystyle\frac{1-\alpha+\alpha\operatorname{var}(N)-\sqrt{(1-\alpha)^2-\operatorname{var}(N)h(\alpha)}}{\operatorname{var}(N)}&\text{if }\alpha\leq\alpha_c,\\[6pt]\infty&\text{if }\alpha>\alpha_c.\end{cases}$$

The conjecture predicts a phase transition at $\alpha_c$: the jump from finite to infinite expected root output coincides with the onset of long-range dependence, above which the occupied cluster of the root appears to occupy a positive fraction of the tree.