Moment lower-bound conjecture for bootstrap percolation on Galton–Watson trees

Let $T_\xi$ be a Galton–Watson tree with offspring distribution $\xi$. Moment lower-bound conjecture. For every $r\geq 2$ and $\alpha\in(0,r-1]$, there exists a constant $c_{r,\alpha}>0$ such that

$$p_c(T_\xi,r)\geq c_{r,\alpha}\left({\mathbb E}(\xi^{1+\alpha})\right)^{-1/\alpha}.$$

This extends the previously established moment lower bound from $\alpha\in(0,1]$ to all $\alpha\leq r-1$; the source gives no resolution.