Conjecture on the extremal process of branching Brownian motion

Let $Xi(t)$ denote the extremal process of branching Brownian motion and let $Pi_r$ be the cluster point process obtained by superimposing independent branching Brownian motions with drift $- √2$ on a Poisson point process with conditional density

$$C Z\bigl(-x\mathrm{e}^{-\sqrt{2}x}\bigr)\,\mathrm{d}x$$

on the negative axis. For a measurable compactly supported function $phi\colon\mathbb{R}\to\mathbb{R}_+$, define

$$\mathrm{e}^{- \psi_r(x)}=E\left[\exp\left(-\sum_{k\leq n(r)}\phi\bigl(x+x_k(r)-\sqrt{2}r\bigr)\right)\right].$$

Extremal-process conjecture. In the limit of large times, the distributions of $Xi(t)$ and $Pi_r$ coincide:

$$\lim_{t\to\infty}\Xi(t)\stackrel{(d)}{=}\lim_{r\to\infty}\Pi_r.$$

In particular,

$$\lim_{t\to\infty}\mathbb{E}\left[\exp\left(-\sum_{k\leq n(t)}\phi\bigl(x_k(t)-m(t)\bigr)\right)\right] =\lim_{r\to\infty}E\left[\exp\left(-CZ\int_{-\infty}^{0}\bigl(1-\mathrm{e}^{-\psi_r(x)}\bigr)\{-x\mathrm{e}^{-\sqrt{2}x}\}\,\mathrm{d}x\right)\right].$$

The conjecture identifies the limiting extremal process with a Poissonian cluster process and would provide a detailed description of its statistics. The preceding discussion explains that existence of the limiting cluster process and its statistical properties are expected to follow from unusually large displacements of branching Brownian motion clusters, but they had not yet been established in this paper.