Let $f: \mathbb{C} \to \mathbb{C}$ and $g: \mathbb{C} \to \mathbb{C}$ be holomorphic functions s.t. $f(0) = 0$ and $g(0)=0$. Let $F: \mathbb{C}^2 \to \mathbb{C}^2$ be defined $F(z,w) = (z+f(w), w)$ and $G: \mathbb{C}^2 \to \mathbb{C}^2$ be defined $G(z,w) = (z, w+g(z))$.
We say that a holomorphic mapping $H: \mathbb{C}^2 \to \mathbb{C}^2$ has a square root if there exists a holomorphic mapping $h: \mathbb{C}^2 \to \mathbb{C}^2$ s.t. $h \circ h = H$.
It is clear that square roots exist for $F$ and $G$. My question is does there exist a square root for $F \circ G$?
Ideas so far: I suspect there does but have limited insight. However, I have one idea. First note that the $n$th root of $F$ exists, and denote it by $F^{\frac{1}{n}}$. Similarly, for $G$. Then I suspect $$\lim_{n \to \infty} (F^{\frac{1}{n}} \circ G^{\frac{1}{n}})^n = F \circ G$$ where the power $n$ denotes the $n$ compositions.
Thoughts and references are appreciated.