Existence conjecture for the planar fixed-diameter truncated-Laplacian problem

Let

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

and let $\operatorname{diam}(\Omega)$ denote the diameter and $\mu_1(\Omega)$ the first eigenvalue of the truncated Laplacian. Fix $d>0$.

Fixed-diameter existence conjecture. The minimization problem

$$\inf\{\mu_1(\Omega)\mid \Omega\in\mathcal{K}_2,\ \operatorname{diam}(\Omega)=d\}$$

admits a solution.

The source studies compactness of minimizing sequences under the diameter constraint and indicates that a minimizer should exist, subject to the regularity issue for eigenfunctions. No resolution is stated.