Very cool. Have done quite some maths in tbe past but never ever heard about that one.

I cannot resist to think about a way to prove it. One can parametrize the shape and formulate it was an optimization problem: find a shape that is maximum so that it fullfills the boundary conditions and can be pulled along a continuous trajectory that itself is unknown. Thats the probably easy part, but how to find that shape by solving it is yet unclear to me. I have other ideas but takes time which I dont have :(