I think the answer is YES .

 Suppose that $d$ is a little square that lies in a unique maximal subrectangle $C$ in the polygon. Then any anti-rectangle contains a unique little square in $C$. We can replace this square by $d$ and still have an anti-rectangle.

Now I think one can show that some corner square must lie in a unque maximal subrectangle. If needs be I can try and write a proof down when I have time.