Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix and $D\in\mathbb{R}^{n\times n}$ be a diagonal positive definite matrix. Let $\mathrm{diag}(X)\in\mathbb{R}^{n\times n}$ denote the diagonal part of a matrix $X\in\mathbb{R}^{n\times n}$.

My question:Does the following inequality hold true $$ \mathrm{diag}\left[(A+D)^{-1}\right] \ge \left[\mathrm{diag}(A)+D\right]^{-1}\ \ ? $$

For the sake of clarity, I describe below a simple example. Consider $$ A=\begin{bmatrix}2 & 1 \\ 1 & 3\end{bmatrix}, \quad D=\begin{bmatrix}0.5 & 0 \\ 0 & 2\end{bmatrix}. $$ We have $$ \mathrm{diag}\left[(A+D)^{-1}\right] = \begin{bmatrix}0.2927 & 0\\0 & 0.1341 \end{bmatrix}, \quad \left[\mathrm{diag}(A)+D\right]^{-1} = \begin{bmatrix}0.1818 & 0 \\0 & 0.0833 \end{bmatrix}, $$ so that $\mathrm{diag}\left[(A+D)^{-1}\right] \ge \left[\mathrm{diag}(A)+D\right]^{-1}$.

(If true, the above inequality perhaps is just a well-know inequality but I wasn't able to find it in any "classical" matrix theory textbooks.)