Consider the classical Julia set $J_c$ associated with $z\mapsto z^2+c$.
Then the image of $J_c$ under the inverse map $z\mapsto \pm \sqrt{z-c}$ lie in $J_c$, since $J_c$ is completely invariant.
Now, let $H_c$ be the convex hull of $J_c$.

**Is it true that the image of $H_c$ under the map $z\mapsto \pm \sqrt{z-c}$ lie in $H_c$?**

I have done some basic computer experiments, and it seem to hold for $c \in [0,1]^2 \subset \mathbb{C}$. Moreover, I suspect that the natural generalization of the statement above might hold for all polynomial maps. However, I have examples with rational maps where the statement is not true.