Mean-regime conjecture for the maximum and extremal process of two-speed branching random walk

Suppose that $\mathcal{L}_1$ and $\mathcal{L}_2$ are non-lattice. Assume that there exist $\theta_1^*,\theta_2^*>0$ such that $\mathcal{L}_1$ and $\mathcal{L}_2$ satisfy assumptions (6), (7), and (8), with $\theta_1^*=\theta_2^*$. Let $\theta:=\theta_1^*$ and

$$m_n:=\kappa_1'(\theta)\lfloor tn\rfloor+\kappa_2'(\theta)(n-\lfloor tn\rfloor)-\frac{3}{2\theta}\log n.$$

Mean-regime conjecture. There exists a constant $C_{\mathrm{m}}>0$ such that, for every $y\in\mathbb{R}$,

$$\lim_{n\to\infty}\mathbb{P}(M_n^{(n)}-m_n\leq y)=\mathbb{E}\left(e^{-C_{\mathrm{m}}Z^{(1)}e^{-\theta y}}\right),$$

and the extremal process

$$\mathcal{E}_n:=\sum_{|u|=n}\delta_{V^{(n)}(u)-m_n}$$

converges in law to a randomly shifted decorated Poisson point process $\mathcal{E}$ with random intensity measure $C_{\mathrm{m}}Z^{(1)}\theta e^{-\theta y}dy$ and decoration $\mathcal{D}^{(2)}$. Here $\mathcal{D}^{(2)}$ is the limit in law of

$$\sum_{|u|=n}\delta_{V_2(u)-\max_{|u|=n}V_2(u)}$$

conditioned on $\{\max_{|u|=n}V_2(u)\geq\kappa_2'(\theta)n\}$. The conjecture concerns the as-yet unstudied mean regime for branching Brownian motion; it predicts both the limiting law of the centered maximum and the limiting extremal process.