 Solution

The unrotated squares have a gap $d>a$, so their facing exclusion layers do not overlap. Rotation can bring them into contact even though their centers remain fixed. The strongest depletion force comes from parallel faces in contact over a long shared length, rather than a corner-to-corner contact.

Let $r=\ell +d$. Rotate both squares through the same angle $\theta$ relative to their original orientation. Their center displacement has a face-normal component $r\cos \theta$ and a tangential component $r\sin \theta$. Face contact requiresThe contacting faces overlap for lengthThese two mirror-related orientations, together with equivalent square rotations, maximize the leading overlap area when $a<d\ll \ell$. A small polymer gap sacrifices overlap width without a compensating leading increase in contact length.