Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$?