In my previous problem, I made a typo. Now I restate it as a new problem.
Let $ \begin{bmatrix} A& B \\ B^* &C \end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$. Is it true that $\quad \sum\limits_{i=1}^k\lambda_i\begin{bmatrix} A& B \\ B^* &C \end{bmatrix}\le \sum\limits_{i=1}^k\left(\lambda_i(A)+\lambda_i(C)\right)\quad, $ where $1\le k\le n$? Here, $\lambda_i(\cdot)$ means the $i$th largest eigenvalue of $\cdot\quad$