Consider the dynamic system $$ \dot{x}=Ax+Bu \\ y=Cx+Du $$ where $ A, B, C, D$ are all real constant matrices. $A$ is square while others may not. Also, assume $A$ is stable (all eigenvalues have negative real parts). The transfer matrix (from $u$ to $y$) in the Laplace domain is given by $$ H(s)=C(sI-A)^{-1}B+D$$ The $\bf{H}_\infty$ norm of the transfer matrix is defined by $$\left\lVert H\right\rVert_\infty\triangleq\sup_{\text{Re}(s)>0}{\sigma_{\text{max}} (H(s))}$$ where $\sigma_{\text{max}}(H(s))$ denotes the maximum singular value of the matrix $H(s)$.
My question is: Why does the $\bf{H}_{\infty}$ norm of $H$ always occur on the imaginary axis, i.e., $$ \left\lVert H\right\rVert_\infty\triangleq\sup_{\text{Re}(s)>0}{\sigma_{\text{max}} (H(s))} = \sup_{\omega\in \mathbb{R}}{\sigma_{\text{max}} (H(j\omega))}$$
(as stated in here)? Is it a general rule for any arbitrary matrix as a function of $s$, or is special to the transfer matrix of a stable state-space system?
If the $\bf{H}_\infty$ norm is an analytic function in $s$ over the RHP, then by the maximum modulus principle plus the fact that $||H(s)||\rightarrow0$ as $\text{Re}(s)\rightarrow \infty$ in our system, we can argue that the supremum will certainly occur on the imaginary axis. However, as far as I know, the maximum singular value may not be a smooth function, not to say analytic (?). Appreciate any comment.