Free-amoeba diversification conjecture

Consider the toy genetics process in which an amoeba has a DNA configuration of length $n$, and each generation produces two offspring through independently activated crossover symbols. Call an amoeba free if its DNA consists of all different bases. Free-amoeba diversification conjecture. There exists a constant $c>0$ such that, starting with any DNA configuration of length $n$, the probability that there is a free amoeba at generation $c\log n$ is at least $1/2$.

The conjecture formalizes the heuristic that the number of distinct bases roughly doubles each generation. It is introduced as a step toward controlling fixed-word evaluations in the iterated wreath products; no proof is given.