Is there an inward (radial) force acting on a magnet inside a magnet?

With a magnet inside another hollow magnet, for example an iron rod in a hollow cylinder, the iron bar placed inside the cylinder with poles attracted to opposite poles of cylinder (likes repel opposites attract), is there also a force along the radius of the cylinder? Per the closed loop rule, the magnetic flux along the sides of the iron rod flows in opposite direction to that of the cylinder. It would seem like the magnetic flux on the inside of the cylinder "squeezes" its way past opposite flux from the iron rod (the iron rod being fixed in place along the longitudinal axis by attraction at poles), compressing it from all directions at once, repelling the iron rod with a vector towards the centre of the radial axis.

The inward (radial) force along the sides of a magnet inside a magnet could be why this piece of metal levitates not only at the center longitudinally but also at the center of the transverse plane.

This radial force could be illustrated in two bar magnets side by side, having slight repulsion along the sides while they attract (with more magnetic force) at the poles.