Conjugate-tuple characterization of Clifford hierarchy gates

Let $d$ be prime, let $n$ be the number of qudits, and let $k$ be a positive integer. A conjugate tuple is a tuple of pairs of gates satisfying the order and commutation relations of the standard Pauli generators; it is $k$-closed when every gate generated by the tuple is in the $k$-th level of the Clifford hierarchy. Up to phase, gates in the $(k+1)$-th level correspond to such tuples.

Conjugate-tuple conjecture. Gates of the $k+1$-th level of the Clifford hierarchy, up to phase, are in bijective correspondence with conjugate tuples of $k$-th level gates.

This is equivalent to asserting that every conjugate tuple of $k$-th level gates is $k$-closed. The claim is motivated by numerical investigations for one qudit; it concerns levels beyond the Clifford group, where $k$-closedness is not automatic and remains open.