Nonattainment conjecture for the planar fixed-perimeter truncated-Laplacian problem

Let

$$\mathcal{K}_2=\{\Omega\subset\mathbb{R}^2\mid \Omega\text{ is open, bounded and convex}\},$$

and let $P(\Omega)$ denote the perimeter and $\mu_1(\Omega)$ the first eigenvalue of the truncated Laplacian. Fix $c>0$.

Fixed-perimeter nonattainment conjecture. The minimization problem

$$\inf\{\mu_1(\Omega)\mid \Omega\in\mathcal{K}_2,\ P(\Omega)=c\}$$

does not admit a solution. A minimizing sequence is given by rectangles of constant perimeter with one side length tending to zero.

The conjecture is motivated by the preceding nonexistence result for the fixed-volume problem. The source gives no resolution, and identifies the proposed degenerating rectangles as a minimizing sequence.