  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 only if $p$ is affinely conjugated to the
Chebyshev polynomial $T_d$ of degree $d$, to $-T_d$ or a monomial $c z^d$ with
$|c|=1$.
