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 $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Does there exist a choice of retraction $\Phi$ such that, for some $n > 0$ and some $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, we have $$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) > 2 \lVert \Phi \rVert$$ where $\mathrm{sdiam}(X) = \lambda_{\max}(X) - \lambda_{\min}(X)$ is the spectral diameter? For instance: in the case $h = k$ (in which $\Phi = \Psi^{-1}$ would be unique), are there $\Psi$, $n>0$, and $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, for which $$\begin{align} \lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert \Phi \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert \Phi \rVert ? \tag{2}\end{align} $$
(This question is a follow-up to a previous question, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above.)