Conjecture on the Neveu CSBP limit in the pulled regime

Let $\rho_1=1+\frac{\pi^2}{4}$, and let $(\overline N_t)_{t\geq 0}$ denote the rescaled particle-number process. Pulled-regime CSBP conjecture. If $\rho<\rho_1$, under suitable assumptions on the initial configurations, the finite-dimensional distributions of

$$(\overline N_{(\log N)^3t})_{t>0}$$

converge, as $N\to\infty$, to those of a 1-stable (Neveu's) CSBP starting from 1. This predicts the limiting population dynamics in the pulled regime, complementing the established weakly pushed regime and leaving the stated initial-configuration assumptions and convergence to be proved.