Constant T-depth rotations using only clean or dirty ancillas (no catalysts)

Show or disprove that constant T-depth single-qubit z-rotations can be implemented using only clean ancillas or dirty ancillas, without employing any catalyst states, within the Clifford+T framework.

Background

The results in the paper rely on catalyst states to achieve constant T-depth for single-qubit z-rotations. Earlier, the authors distinguish among dirty ancillas, clean ancillas, and catalyst states, with catalysts being the strongest resource considered.

They explicitly pose whether the same constant T-depth capability can be achieved without catalysts, using only clean or dirty ancillas, which would strengthen the resource assumptions of the construction.