Does MDF have that much damping? It certainly has a slight flexibility but I wouldn't think it has much way to convert the energy, just store it. Obviously bracing to the opposite panel will help huge amounts as they will be out of phase and cancel each other.

I would think, and certainly like to try it one day, that something like 2 sheets of 1/2" aluminium honeycomb with a layer of latex painted on and pressed between them would do a great job. Very light weight, incredibly stiff and a very lossy self damping. Go to the effort of bracing it with aluminium as well, and it should make for quite a nice box with a resonance well out of any bass or even mid-bass drivers range. The problem is having the guts to try it out!