Let $\mathbf A=\left[\begin{matrix}\mathbf A_{11}&\mathbf A_{12}\\ \mathbf A_{21}&\mathbf A_{22}\end{matrix}\right]$ be a positive semi-definite matrix, $\mathbf A_{ij}\in\mathbb C^{n\times n}$ and $rank(\mathbf A)=n$.

**Prove or disprove** that
$$
\lambda_{max}(\mathbf A) + \lambda_{max}(\mathbf A_{11}+\mathbf A_{22}) \leq
\lambda_{max}\left(\left[\begin{matrix}tr(\mathbf A_{11})&tr(\mathbf A_{12})\\ tr(\mathbf A_{21})&tr(\mathbf A_{22})\end{matrix}\right]\right) + tr(\mathbf A_{11}+\mathbf A_{22}),
$$
where $\lambda_{max}$ denotes the largest eigenvalue and $tr$ denotes the trace.

I tried small values of $n$. For $n=1$ we have "$=$" (the same terms in the LHS and RHS). I have a rather long proof for $n=2$ (in my proof I also replaced $\mathbb C$ by $\mathbb R$).

It seems that some results on (completely) positive maps are somehow related to the inequality ( indeed, all the maps $$ \mathbf A\rightarrow \mathbf A_{11}+\mathbf A_{22},\qquad \mathbf A\rightarrow \left[\begin{matrix}tr(\mathbf A_{11})&tr(\mathbf A_{12})\\ tr(\mathbf A_{21})&tr(\mathbf A_{22})\end{matrix}\right],\qquad \mathbf A\rightarrow tr(\mathbf A_{11}+\mathbf A_{22}) $$ are positive). By Theorem 2.3.7 (The Russo-Dye Theorem) from the book R. Bhatia, Positive definite matrices, $$ \lambda_{max}\left(\left[\begin{matrix}tr(\mathbf A_{11})&tr(\mathbf A_{12})\\ tr(\mathbf A_{21})&tr(\mathbf A_{22})\end{matrix}\right]\right)\leq n\lambda_{max}(\mathbf A), $$ which is useless since we need inequality of the type $\lambda_{max}(\mathbf A)\leq\dots$