The degree-sharp polynomial iterate multiplier conjecture

Let $p$ be a polynomial of degree $d\geq 2$, and let $n\geq 2$. Degree-sharp multiplier conjecture. Then $p^n$ has a fixed point $\xi$ satisfying

$$|(p^n)'(\xi)|\geq d^n.$$

The source states that this is stronger than the preceding polynomial multiplier conjecture. The general assertion is presented as an open conjecture and would give a degree-dependent sharp lower bound for multipliers of fixed points of iterates.