Nonattainment and collapsing rectangles for the infimum of the Steklov–Neumann functional

Let $F(\Omega)$ be the functional introduced in the paper, and consider bounded, convex, open sets $\Omega\subset\mathbb{R}^2$. The associated minimization problem is

$$\inf\{F(\Omega)\mid \Omega\subset\mathbb{R}^2\ \text{bounded, convex and open}\}.$$

Nonattainment conjecture. The displayed minimization problem has no solution. In particular, every minimizing sequence $\Omega_\epsilon$ must be of the form of collapsing rectangles.

The claim concerns the structure of minimizing sequences: the paper observes that collapsing rectangles approach the infimum numerically, while no bounded convex open domain is expected to attain it.