Informal stable-CSBP and genealogy convergence conjecture for spatial branching processes

Let $\tilde h^\tinfty$ denote the invariant spatial distribution of particles, and suppose that, for some slowly varying function $\ell$ and $\alpha\backslashin(1,2)$, condition

$$\tilde h_{\infty}( h_{\infty}^{-1} (x,\infty)) \sim \frac{\ell(y)}{y^\alpha}$$

holds as $y\to\infty$. Stable-CSBP and genealogy conjecture. Assume that this condition holds. Then the process describing the total number of particles and the genealogy at a given time become indistinguishable from an $\alpha$-stable CSBP and its genealogical structure. Further, the particles are asymptotically distributed as independent identically distributed random variables with distribution $\tilde h^\infty$. This is an informal heuristic for the joint scaling limit of the population and genealogy; the statement is presented without a precise topology or a resolution status in the source.