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](https://mathoverflow.net/questions/283787/completely-bounded-norm-for-unital-maps-with-completely-positive-sections), in which it was established that $\Psi$ may have retractions which do *not* have this property.)