Matrix Submodular Inequality Given $a,b,x > 0$ I know following the submodularity property holds:
\begin{align}
\frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x}
\end{align}
My question is, does this property hold for matrices? Precisely, for $A,B,X \succ 0$ is it the case that:
\begin{align}
A^{-1} - (A+X)^{-1} \succeq (A+B)^{-1} - (A+B+X)^{-1}
\end{align}
By '$\succeq$' I mean that if $A \succeq B$ then $A−B \succeq 0$, or, is positive semi-definite.

Consider 
$$ A = \pmatrix{1 & 0\cr 0 & 1\cr},\ X = \pmatrix{1 & 0\cr 0 & 0\cr},\ B = \pmatrix{1 & 1\cr 1 & 1\cr}$$
$$ \eqalign{A^{-1} &- (A+X)^{-1} = \pmatrix{1/2 & 0\cr 0 & 0\cr}\cr &\not\succeq 
(A+B)^{-1} - (A+X+B)^{-1} = \pmatrix{4/15 & -2/15 \cr -2/15 & 1/15}}$$
Yes, I know $B$ and $X$ are positive semidefinite rather than positive definite; add $\epsilon I$ for $\epsilon $ sufficiently small.
