Conjecture on the limiting empirical mass distribution of red leaves

Fix a critical initial law with $(p(0),\lambda(0))\in\mathscr{C}$. Let $\mathfrak{N}^t_t$ be the set of red leaves at time $t$, let $\mathfrak{X}^t_t(u)$ be the mass of $u\in\mathfrak{N}^t_t$, and let $N_t=\#\mathfrak{N}^t_t$. For positive real numbers $x_t$ satisfying $x_t/t\to x\in\mathbb{R}_+$ as $t\to\infty$, consider the empirical random measure

$$\frac{1}{N_t}\sum_{u\in\mathfrak{N}^t_t}\delta_{\mathfrak{X}^t_t(u)}.$$

Red-leaf empirical-measure conjecture. Under $\mathbb{P}_{x_t}$, this random measure converges in law for the topology of vague convergence, and its limiting law does not depend on $x$. This concerns the typical mass of a red leaf in the critical Derrida–Retaux model; the source explicitly presents it as an unanswered further question and does not identify the limiting measure.