The generalized semi-Clifford conjecture for the qubit Clifford hierarchy

A gate in $U(2^n)$ is generalized semi-Clifford if it can be expressed as

$$U=C_1P\prod_j\exp\left(i\alpha_j Z_j\right)C_2,$$

where $C_1,C_2$ are $n$-qubit Clifford gates, $P$ is a permutation matrix in $U(2^n)$, and the $Z_j$ are $Z$ Pauli strings. The generalized semi-Clifford conjecture. Every element of the qubit Clifford hierarchy is generalized semi-Clifford. The table of known results shows that this holds in the cases established so far, while the assertion for all levels and numbers of qubits remains open.