Optimal T-count for variable-angle rotations at constant T-depth (catalytic)

Determine whether there exists a catalytic Clifford+T circuit with constant T-depth that, for variable single-qubit z-rotation angles, achieves an epsilon-approximation using T-count O(log(1/epsilon)).

Background

The paper shows that single-qubit z-rotations can be implemented with T-depth 3 using a catalyst state. For a fixed rotation angle, both the catalyst size and T-count scale as O(log(1/epsilon)). However, when the rotation angle varies, the presented constructions increase the resource scaling to O(log^2(1/epsilon)).

The authors explicitly ask whether the variable-angle case can match the fixed-angle scaling while retaining constant T-depth in the catalytic setting, i.e., whether the T-count can be reduced to O(log(1/epsilon)) for variable-angle rotations.