The differential Galois group of the maximal prounipotent extension is free

Let $F$ be a characteristic zero differential field with algebraically closed constant field, and consider the compositum $F_u$ of all Picard–Vessiot extensions of $F$ with unipotent differential Galois group. We prove that the group of $F$ differential automorphisms of $F_u$ is a free prounipotent group.