The commuting-Toffoli characterization of third-level permutation gates

Let $\mathcal{CH}_k$ denote the $k$th level of the qubit Clifford hierarchy, and let a permutation gate be a unitary that permutes the computational-basis states. A Clifford permutation is a permutation gate belonging to the Clifford group, and a Toffoli gate is a doubly controlled bit flip, denoted $C^2(X)$. The commuting-Toffoli conjecture. A permutation is in $\mathcal{CH}_3$ if and only if it can be written as a circuit of commuting Toffoli gates, possibly preceded and followed by Clifford permutations. The forward implication is open; the reverse implication follows from the results described in the paper.