The low-density limit of coalescing branching random walk

This is one of the "something like this must be true" situations. The issue is to find a reasonably "clean" proof technique that can be adapted to both models below, and hopefully to other variants too.

For concreteness take the lattice $Z^3$ -- what's important is random walk is transient.

Model 1. Consider a model BCRW of branching coalescing random walk. To be definite (though the point is that the details shouldn't matter) take particles behaving as:
(i) continuous-time simple RW
(ii) splitting at rate $\beta$, a daughter particle placed at an adjacent site
(iii) two particles coalesce into one if they meet.
Such a process has a (translation-invariant) stationary distribution for which the mean number of particles per site (intensity) equals some $p(\beta)$. See this Athreya-Swart paper for discussion. There seems no simple exact formula for the complete stationary distribution, but one can calculate $p(\beta) \sim c \beta$ as $\beta \to 0$.

Model 2. This is the constrained Ising model from page 16-17 of The constrained Ising model as an algorithm for storage in dynamic graphs but now with $Z^3$ instead of a finite graph. The model has a parameter $p$ and in the stationary distribution, site are occupied as an i.i.d. Bernoulli(p) process. As commented in the linked material, this process evolves in a way that emulates some variant of CBRW.

Problem. The issue is to show that, for Models 1 and 2 and related models where there is a stationary distribution, as the intensity (which depends on model parameters) tends to 0, the suitable time-rescaled process in which we track the ``lineages" (splits and merges) but not the spatial positions, of the particles, converges in distribution to the following simple limit process.

A limit process. The limit process is constructed w.r.t. a reference particle at time 0. This has a lineage from the infinite past to the infinite future; ``splits" and ``merges" with some other lineage both occur at times of Poisson(1) point processes. A split [merge] creates a new lineage running froward [backward], and recursively such a lineage has its own splits and merges, each creating a new lineage.

In the conjectured convergence result we need to overlook "jitter", the times where a daughter particle soon coalesces with its parent, which we do by asking only for convergence of finite dimensional distributions (in time).