Boundary extremal-process conjecture between $\mathcal{C}_I$ and $\mathcal{C}_{III}$

Let $\beta>1$ and $\sigma^2=\frac{\beta}{2\beta-1}$, and let $\mathcal{N}_t^2$ be the set of type-$2$ particles at time $t$, with positions $X_u(t)$. Write $Z^{(1)}_\infty$ for the type-$1$ derivative martingale limit, and let $\operatorname{DPPP}(\mu,\mathfrak{D})$ denote a decorated Poisson point process with intensity measure $\mu$ and decoration point measure $\mathfrak{D}$. Boundary extremal-process conjecture. There exist $c>0$ and a random decoration point measure $\widetilde{\mathfrak{D}}$ such that

$$\sum_{u \in \mathcal{N}_t^2} \delta_{X_u(t) - \sqrt{2\beta\sigma^2}t + \frac{1}{\sqrt{2\beta/\sigma^2}} \log t}$$

converges in distribution to

$$\operatorname{DPPP}\left(c Z^{(1)}_\infty e^{-\sqrt{2\beta/\sigma^2}x}\,\mathrm{d}x,\widetilde{\mathfrak{D}}\right).$$

Particles of type $2$ contributing to the extremal process are expected to satisfy $t-T(u)=O(t^{1/2})$, so the intensity should be driven by the type-$1$ derivative martingale, while the decoration should be the extremal process of a type-$2$ BBM conditioned to travel at speed $\sqrt{2\beta\sigma^2}>\sqrt{2}$. The conjecture concerns the boundary between the two established parameter regions and remains open.