The inverse-closure conjecture for permutation gates in the Clifford hierarchy

Let $\mathcal{CH}_k$ denote the $k$th level of the qubit Clifford hierarchy, and let $\pi$ be a permutation gate on the computational-basis states. The inverse-closure conjecture. If $\pi$ is in $\mathcal{CH}_k$, then its adjoint $\pi^\dagger$ is also in $\mathcal{CH}_k$. This is a natural closure question for permutation gates in the hierarchy, and its resolution is not established by the supplied text.