Conjecture on the Feller diffusion limit in the fully pushed regime

Let $\rho_2$ be the threshold separating the weakly pushed and fully pushed regimes, and let $(\overline N_t)_{t\geq 0}$ denote the rescaled particle-number process. Fully pushed Feller diffusion conjecture. If $\rho>\rho_2$, under suitable assumptions on the initial configurations, the finite-dimensional distributions of

$$(\overline N_{Nt})_{t>0}$$

converge, as $N\to\infty$, to those of a Feller diffusion starting from 1. This is the expected population-level limit in the fully pushed regime; the paper identifies it as a direction for future work, so the convergence remains open.