Example: Nonconstant need not mean large leakage

For a qubit define

For $0<\theta <\pi /2$, the complement is nonconstant, but

The accessible channel tends to identity as $\theta \to 0$.

Indeed, the complement is a measurement in the logical basis followed by preparation of two pure states whose trace distance is $\sin \theta$. Their average gives the stated upper bound; the triangle inequality applied to the two outputs gives the matching lower bound. Appending arbitrarily large fixed residual spectators changes neither conclusion.