I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p_i$ of a child surviving; so each node has 2 children with probability $p_i^2$, 1 child with probability $2 p_i (1-p_i)$, and zero children with probability $(1-p_i)^2$.

I want to show that the process survives to time $T$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous. The number of expected survivors at level $i$ is $\mu_i = 2^i p_1 \dots p_i$, which I can compute. Is there some criterion of the form $\forall i, \mu_i \geq \text{poly}(n)$ that suffices to show survival?

I am sure that this is well-known, but I have not been able to find any simple description.

Thanks!