Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions. Does $\Psi$ admit a retraction $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$, such that $\lVert \Phi \rVert = \lVert \Phi \rVert_{\mathrm{cb}}$?

(This question is a follow-up to a previous question, in which it was established that $\Psi$ may have retractions which do not have this property.)