This question is related to https://mathoverflow.net/questions/70689/ask-some-matrix-eigenvalue-inequalities

Let $\begin{bmatrix}
A& B  \\\\ B^*  &A
\end{bmatrix}$ be positive semidefinite. Is it true $\lambda_i^{1/2}(B^*B)\le \lambda_i(A)$?      Here, $λ_i(⋅)$ means the ith largest eigenvalue of ⋅.