Incidentally, a flush door is a ready made torsion box, and might make a good assembly table for the lazy. Over here they start from £24, but the biggest size is too small for full sheet. I would reinforce with a second MDF skin too, and maybe edge it for durability. I have no idea how flat and true they are off the shelf.
You can also get a ply fire door at up to 8×4 here for £80. They are solid core and hardwood faced. Again not sure how flat, but clearly better built in rigidity than raw sheet material. Perhaps you could add something to the edges to support your motion?