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}^n\lambda_i\begin{bmatrix} A& B \\ B^* &C \end{bmatrix}\le \sum\limits_{i=1}^n\left(\lambda_i(A)+\lambda_i(C)\right)\quad? $
Here, $\lambda_i(\cdot)$ means the $i$th largest eigenvalue of $\cdot\quad$