Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. 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 $\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)