Geometric limit conjecture for the largest family sizes in the critical case

Let $Z_p$ be a critical clonal process in a general supercritical splitting tree, with finite second moment

$$\sigma^2:=\int_0^\infty \Lambda(\mathrm{d}u)u^2$$

and let $r$ denote the Malthusian growth rate. Define

$$x_t:=\frac{2}{\sigma^2}(rt^2-t\log t).$$

Geometric limit conjecture. As $t\to\infty$, on the survival event, $L_t(x_t)$ converges in distribution to a non-degenerate geometric random variable. This conjecture concerns the still-open problem of determining the order of magnitude and limiting behaviour of the largest families in the critical case; the surrounding discussion notes that analogous results are known under additional assumptions, but the size problem itself remains open.