A Galton-Watson tree is the family tree of a Galton-Watson process. Let $T_n$ denote a Galton-Watson tree conditioned on total population size $n$. The time of extinction is its height $H(T_n)$ and its diameter $D(T_n)$ is the length of a longest path in the tree. It is known that for certain offspring distributions the ratio $\mathbb{E}[D(T_n)] / \mathbb{E}[H(T_n)]$ converges to $4/3$. The same property is expected to hold in a more general context, as argued heuristically by Aldous in the paper "The continuum random tree II: an overview" (1991).

I would like to know if a formal proof of this is known in the case of an arbitrary offspring distribution with expected value 1 and finite nonzero variance.