The $L^\infty$ maximizer conjecture for the plate load problem

Let $\Omega$ be the rectangular plate, let $D\subset\Omega$ be the relevant hinged set, and let $p=\infty$. Consider the maximization problem over admissible loads in the corresponding $L^p$ class. An odd function satisfies $f(x,y)=-f(x,-y)$.

$L^\infty$ maximizer conjecture. For every $D\subset\Omega$, the unique maximizers of the problem are the odd function

$$f(x,y)=\frac{y}{|y|},\qquad y\neq 0,$$

and its opposite $-f$.

The conjecture concerns the maximization problem in the endpoint space $L^\infty$; the authors contrast it with a symmetry property proved in the paper and leave the proposed uniqueness assertion unresolved.