Minimum dimensions for transpose antidegradability

Determine whether transpose-antidegradable quantum channels exist with input, output, and environment dimensions smaller than the currently known triple (4,4,6), given that the minimum possible dimension triples are (2,4,5), (3,3,4), and (4,2,5).

Background

The paper constructs a transpose-antidegradable two-qubit Pauli channel with input, output, and environment dimensions (4,4,6). It notes that this improves on a contemporaneous construction with dimensions (4,4,8), but does not establish whether still smaller realizations are possible.

The unresolved issue concerns the dimensional resources required for transpose antidegradability. The cited candidate triples specify lower-dimensional configurations whose feasibility remains unsettled.