Given the invertible symmetric block diagonal matrices $S = S_1\oplus S_2$ and $dS = dS_1 \oplus dS_2$, I want to find the symmetric matrix X with blocks $$ X = \left[ \begin{array}[cc] ~X_1 & X_{12}\\ X_{12}^T & X_2 \end{array} \right] $$ that maximises the following objective (which is related to the Fisher information of a Gaussian distribution, if you are familiar with that) $$ F = {\rm trace} [(dS(X+S)^{-1})^2], $$ subject to $$ X + i\Omega \succcurlyeq 0, {\hspace .2cm }\text{with } \Omega = \omega\oplus\omega, {\hspace .2cm} \text{and } \omega = \left[ \begin{array}[cc] ~0 & 1\\ -1 & 0 \end{array} \right]. $$ The known matrix $S$ also satisfies this same criterion, i.e., $S+i\Omega \succcurlyeq 0$. For our purpose, we can also assume $S_i$ and $dS_i$ are diagonal for $i \in \{1,2\}$.
I think in general this can be a difficult optimisation problem. I have the conjecture that the optimal $X$ should have off-diagonal blocks vanish, i.e., $X_{12}^{\rm opt} = 0$. Unfortunately, I am stock in the matrix inversion and I cannot prove this. I would appreciate it if anyone can give me some hint. Alternatively, if one could give a counter example for which the optimal $X_{12}^{\rm opt}\neq 0$ it would be great.
