After you place the cube in the box you can easily see 2 pieces of card on either side of the box where there was just one before inserting the cube, showing that the sides of the cube stayed on top, enabling the rod to be pushed through. Might be able to conceal that if there were a piece of decorative paper at the top of the box overhanging by roughly the thickness of the card that stays at the top as cube is pushed in, and making sure the card that stays at top goes into the box so it will be just past the overhang so it will be concealed.
I hired a magician for my son's 11th birthday. After he saw the trick, he asked magician to lift the box out by holding the front and back sides. The magician said he needed to move on to the next trick because he didn't have enough time.