Loewner ordering of matrix products

Question

Let $B \geq A \geq 0$ and $M \geq 0$ be real symmetric matrices. Is it true that

$B M B \geq A M A$, i.e., $B M B - A M A \geq 0$?

Answer 1

No. It is known, for instance, that $B\ge A\ge0$ does not imply $B^2\ge A^2$. Now pick such a pair of matrices $A$ and $B$ and set $M=I$.