Let $G_\mathrm{ss}(\omega)= \begin{bmatrix} \Re\{\underline{G}(j\omega)\} & \Im\{\underline{G}(j\omega)\} \\ -\Im\{\underline{G}(j\omega)\} & \Re\{\underline{G}(j\omega)\} \end{bmatrix}$ be the steady-state system matrix of a MIMO system $G(s)$ with arbitrary number of inputs and outputs.
Is it possible to proove that following always holds for a series connection $G(s) = G_1(s) \, G_2(s) \, G_3(s)$ of such systems of arbitrary (but for sure compatible) size:
$ G_\mathrm{ss}^+(\omega) = (G_{1,ss}(\omega) \, G_{2,ss}(\omega) \, G_{3,ss}(\omega))^+ = G_{3,ss}^+(\omega) \, G_{2,ss}^+(\omega) \, G_{1,ss}^+(\omega) $
$(\cdot)^+$ is the Moore Penrose inverse.