Max norm of $\mathbb C$-linear extension of $\mathbb R$-linear map on Hermitian matrices

This question is an extension of Norms of linear maps.

Let $\Phi:H_n(\mathbb C)\to H_n(\mathbb C)$ be a real linear map of norm $1$ on Hermitian $n$-by-$n$ matrices ($n>1$) with the operator norm induced by the Euclidean norm on $\mathbb C^n$. Let $\overset{\sim}\Phi:M_n(\mathbb C)\to M_n(\mathbb C)$ be the $\mathbb C$-linear extension of $\Phi$ defined by $$\overset{\sim}\Phi(A)=\Phi\left({A+A^* \over 2}\right) + i\Phi\left({A-A^* \over 2i}\right).$$

What is the largest possible norm of $\overset{\sim}\Phi$?

Remarks:

• The answer may depend on $n$. Perhaps the bounds continue to increase with $n$, so an ideal solution may find the answer as a function of $n$, if there is not a unique largest possible norm that applies for all $n$.
• An upper bound of $2$ follows from \begin{align*}\|\overset{\sim}\Phi(A)\|&=\|\Phi(\mathrm{Re}(A))+i\Phi(\mathrm{Im}(A))\|\\ &\leq \|\Phi(\mathrm{Re}(A))\|+\|\Phi(\mathrm{Im}(A))\|\\ &\leq \|\mathrm{Re}(A)\|+\|\mathrm{Im}(A)\|\\ &\leq 2\|A\|.\end{align*}
• A lower bound of $\sqrt2$ follows from an example where $n=2$ and $\|\overset{\sim}\Phi\|=\sqrt2$ given at the above linked question.
• Useful partial answers could involve just answering the question for a few small $n$ even if the general answer isn't revealed, or just finding an example with $\|\overset{\sim}\Phi\| >\sqrt2$ if one exists.