Example: The missing coherences are an operational difference

The channels $\mathcal{A}$ and $\mathcal{A}\Delta$ agree on the two computational-basis inputs and have the same computational population output for every input. On the input $|+⟩⟨+|$, however, their outputs differ by

so their output trace distance is $k/2$. At the frozen parameters this is $1/(2\sqrt{2})>0$.

Thus the population-transition records alone do not certify the full GAD channel. The Holevo calculation uses the coherence term as well.