Long-term behavior dichotomy for branching diffusion with interactions

Consider the interacting branching diffusion in the case $b>0$ and $b+\gamma<0$, with $Z_n$ denoting its particle-counting measure at time $n$ and $d$ the spatial dimension. The process has an outward Ornstein–Uhlenbeck relative motion, while its center of mass converges to $0$ as time tends to infinity.

Long-term behavior dichotomy. The following dichotomy holds:

1. If $\frac{\log 2}{d}\leq |b+\gamma|$, then $Z$ suffers local extinction.
2. If $\frac{\log 2}{d}>|b+\gamma|$, then

$$2^{-n}e^{d|b+\gamma|n}Z_n(\mathrm{d}y)\overset{v}{\Rightarrow}\mathrm{d}y.$$

This conjecture describes the competition between branching and the outward spreading of the relative system. The source presents it as analogous to a conjecture for a related model; no resolution is supplied here.