The sharp normality conjecture for iterates

Let $D\subset\mathbb C$ be a domain, let $n\geq 2$, and let $\varepsilon>0$. Let $\mathcal F$ be the family of all holomorphic functions $f:D\to\mathbb C$ such that

$$|(f^n)'(\xi)|\leq 2^n-\varepsilon$$

for every fixed point $\xi$ of $f^n$. Sharp normality conjecture. Then $\mathcal F$ is normal in $D$. The conjecture is known when $n=2$, while the general case remains open; it predicts the sharp threshold below which the derivative bounds in the normality theorem force normality.