The $3$-closure of a solvable permutation group is solvable

Let $m$ be a positive integer and let $Omega$ be a finite set. The $m$-closure of $Gleqoperatorname{Sym}(Omega)$ is the largest permutation group on $Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $Omega^m$. The $1$-closure and $2$-closure of a solvable permutation group need not be solvable. We prove that the $m$-closure of a solvable permutation group is always solvable for $mgeq3$.