Extremal-process moderate-deviation conjecture for branching random walks

Assume the branching random walk satisfies the supercriticality, boundary-case, Gaussianity, and peeling-lemma assumptions, and is non-lattice. Let $\phi\colon\mathbb{R}\to\mathbb{R}$ be continuous with compact support, and let $(x_n)$ satisfy

$$x_n\to\infty,\qquad x_n=O(\sqrt{n})$$

as $n\to\infty$. Write $m_n$ for the centering sequence, let $V(u)$ denote the position of an individual $u$ at generation $n$, and let $\sigma$ be the variance parameter. Define

$$C_{\mathrm{NL}}(\phi)=\frac{1}{\sigma^3}\sqrt{\frac{2}{\pi}}\lim_{\ell\to\infty}\int_0^\infty y\mathrm{e}^{y}\,\mathbb{E}\mathopen{}\mathclose\bgroup\originalleft[1-\mathrm{e}^{-\sum_{|u|=\ell}\phi(V(u)-y)}\aftergroup\egroup\originalright]\,\mathrm{d}y.$$

Extremal-process moderate-deviation conjecture. As $n\to\infty$,

$$\mathbb{E}\mathopen{}\mathclose\bgroup\originalleft[1-\mathrm{e}^{-\sum_{|u|=n}\phi(V(u)-m_n-x_n)}\aftergroup\egroup\originalright]\sim C_{\mathrm{NL}}(\phi,a)x_n\mathrm{e}^{-x_n-x_n^2/(2n\sigma^2)}.$$

This extends the upper moderate-deviation asymptotic for the maximum to Laplace functionals of the branching random walk viewed from its extremal position, thereby connecting moderate deviations with the limiting extremal process. The supplied statement is presented as a conjecture, but its resolution is not specified in the source material.