Consider a continuous time doubling branching random walk on the nonnegative integers. It starts with a particle at $0$. It dies after producing two offspring at $1$, each after an independent exponential$(1)$-distributed amount of time. In the same fashion, a particle born at $k$ will die upon producing two offspring at $k+1$ after, each after an independent exponential$(1)$-distributed amount of time. Suppose the last of the $2^n$ particles born at $n$ occurs at time $T$. This is known as the "Last-Birth Problem" and is fairly well studied. We are curious how many particles reach $n$ just before $T$.

To make this concrete let $X_n$ be the number of particles that reach $n$ at times in $[T-1,T)$. Is there a function $f(n)\to \infty$ such that $P(X_n > f(n)) \to 1$?

Perhaps this is false and only $O(1)$ particles arrive in the final second. However, there may be a huge wave of particles just in front of the last one. Potentially, we have $f(n) = 2^n / C$ for some $C$, but this could be wishful thinking.