Gamma-shape conjecture for the condensation wave

Let $\Xi_t(f,g)$ denote the mass distribution of families with fitness in the interval determined by $f$ and $g$, and let $\alpha$, $\beta$, and $\gamma$ be the model parameters. Under assumption (as), for $x\geq 0$, Gamma-shape conjecture.

$$\lim_{t\to\infty} \Xi_t\left(1-\frac{x}{t},1\right)=\frac{\omega(\beta,\gamma)}{\Gamma(\alpha+1)}\int_0^x y^{\alpha}e^{-y}\,dy,$$

in probability. Thus the condensation wave should have the shape of a Gamma distribution with shape parameter $1+\alpha$.

This conjecture proposes a universal limiting shape for the random mass distribution near the maximal fitness in the condensation regime, extending the Gamma-shaped behaviour found in related growth models. The paper presents it as an open question for the model considered here.