Inequality between inverses of positive (semi)definite matrices

Question

I know that the following relationship holds: $A\succeq B\succ 0\Rightarrow A^{-1}\preceq B^{-1}$. Is there a similar relationship which characterizes the case when $A\succeq B$, $A\succ 0$ and $B\succeq 0$ (in particular, B is not invertible)? Something like $A\succeq B,A\succ 0, B\succeq 0\Rightarrow A^{-1}\preceq B^\dagger$ where $B^\dagger$ is the Moore–Penrose pseudoinverse.

Answer 1

Doubtful. Let $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and let $B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$. Then $A \succeq B$, and $A=A^{-1}$, while $B^\dagger = B$. However, $A^{-1} \not \preceq B^\dagger$.