The convergence conjecture for the branching-random-walk maximum

Let $B_n$ be the maximum at generation $n$ in the branching random walk considered in the paper, and let $b_n$ be its deterministic centering term. Branching-random-walk convergence conjecture. There is a random variable $X$ with continuous distribution such that

$$B_n-b_n\to X\qquad\text{as }n\to\infty.$$

The paper notes that convergence in probability of this centered maximum is known under certain displacement-law conditions, but is not known for the model arising from Pratt trees.