Goldschmidt and Przykucki's conjecture on the mean number of cars

Let $N$ be the offspring distribution of a critical, non-degenerate Galton--Watson tree, let $P$ be the common distribution of the i.i.d. numbers of cars initially arriving at vertices, and let $X$ be the number of cars arriving at the root. Assume that ${\mathrm{var}(N)}\leq 1$, that $P$ is stochastically increasing in $\alpha={\mathbb{E}[P]}$, and that ${\mathrm{var}(P)}<\infty$ for all $\alpha\geq 0$. Define

$$\nu(\alpha)={\mathbb{E}[P(P-1)]}$$

and

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

Goldschmidt and Przykucki's conjecture. The expected number of cars arriving at the root is

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

This conjecture seeks a universal formula for the mean root load and the critical point across critical Galton--Watson offspring distributions with variance at most one and stochastically increasing arrival distributions. The surrounding discussion reports that analogous formulas and discontinuous phase transitions are known in several specific models, while the general assertion remains unresolved.