What everyone in this thread is saying is that due to the position of the rod (BELOW the mass at the center), and the fact that the rod is massless, the rod is not producing any kind of force. This diagram at all points is equivalent to the mass at the center "floating" in midair.
You may be moving the rod, but no work is being done. The rod is massless and does not contribute to the energy. Only the central mass has mass and it isn't moving.
I don't see how this is paradoxical. The mass never moved from the center, and therefore did no work. It is still located at mgr. I think you are confusing the path taken by the mass at the center with the path taken by the rod.
If the rod were not mass-less, then work would have been done...
Think of potential as the energy required to bring a charge from infinitely far away to the point in question. The closer you get to the point in question (say a positive charge), the higher your potential. Now think of the potential as a hilltop. At infinity it is flat, but as you get closer to...