Could propeller wash provide sufficient lift to take off - even in theory? I know this seems a crazy thing to ask but it is not the same as lifting oneself by the bootstraps. If you try to lift yourself that way, there are two equal and opposite forces, yet there are experiments with muscle powered planes that work.
So, provided the plane was stabilised somehow, and the wings were designed for the purpose, could propellor wash theoretically lift the plane? If not why not?

Note
I understand that a normal plane would start to move forward. Please assume that the plane is restrained from the rear by a cable and that the experiment takes place in the absence of wind.
Yes.  The wing doesn't care what is causing air to flow past it.  Headwind, propwash, 747 wake, gopher sneezes.
If the airplane is restrained from forward motion, and the propwash over most of the wing is fast enough, let's say more than the airplane's stall speed, then the wing must generate enough lift to support the airplane, causing it to take off.
It's not so much a matter of designing the wing specially, as making the airplane tolerate the many large props needed, and the weight of the engines.  That much power is too much for efficient cruise.  But it's easy to implement with a one pound model airplane.