The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ indicates the Schatten-1 norm (also known as the trace norm). Does anyone know if the following extension might be true, or if there's an obvious counter-example?

I'm wondering if there is some constant $C$ such that for positive semidefinite operators (say finite dimensional) $A, B$, we have $$ \mathrm{Tr}((A - B)X^2 (A - B)) \leq C \| XA^2 X - X B^2 X \|_1 $$ where $X = \sqrt{A^2 + B^2}$ (the square root is uniquely defined because $A^2 + B^2$ is positive semidefinite).

Note that if $X$ is an arbitrary positive operator, then this is false. However, if $X$ is related to $A$ and $B$ in this nice way, could this be true (or is anything "like" this that is true)?

Thanks!