Tail asymptotics for the total progeny of the critical killed branching random walk

We consider a branching random walk on $\mathbb{R}$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that the tail distribution of $Z$ displays a typical behaviour in $(n\ln^2(n))^{-1}$, which confirms the prediction of Addario-Berry and Broutin.