On the maximum of a critical branching process in a random environment

Let $\{\xi_n\}$ be a critical branching process in a random environment with linear-fractional generating functions. We demonstrate that, under some conditions, as $x\to\infty$,
$$ \mathsf P\Bigl(\sup_n\xi_n>x\Bigr)\sim \frac{c_0}{\ln x},\qquad \mathsf P\biggl(\sum_{n=0}^\infty\xi_n>x\biggr)\sim \frac{c_0}{\ln x}, $$
where $c_0$ is a positive constant.