Convex hulls of polynomial Julia sets

We prove P. Alexandersson’s conjecture that for every complex polynomial $p$ of degree $d \geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ satisfies $p^{-1}(H_p) \subset H_p$. We further prove that the equality $p^{-1}(H_p) = H_p$ is achieved if and only if $p$ is affinely conjugated to the Chebyshev polynomial $T_d$ of degree $d$, to $-T_d$, or to a monomial $c z^d$ with $|c|=1$.