I have three matrices $A \in \mathbb R^{n \times n}$, $B \in \mathbb R^{n \times n}$, and $X \in \mathbb R^{n \times n}$.

Suppose that $A$ is singular, $B = B^\top > 0$ and $X = X^\top > 0$.

Then, does the following inequality true?

$A^\top (A X^{-1} A^\top + B)^{-1} A \leq X$

My approach was decomposing $A$ into singular and non-singular part but, it was still unsuccessful...