Reuleaux-triangle minimizer conjecture for the planar fixed-diameter problem

Let $\mathcal{K}_2=\{\Omega\subset\mathbb{R}^2\mid \Omega\text{ is open, bounded and convex}\}$, let $\operatorname{diam}(\Omega)$ denote the diameter, and let $\mu_1(\Omega)$ be the first eigenvalue of the truncated Laplacian. For $d>0$, consider

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

Let a Reuleaux triangle mean the constant-width convex domain generated by an equilateral triangle.

Reuleaux-triangle minimizer conjecture. The Reuleaux triangle is a minimizer for the fixed-diameter problem.

The conjecture is motivated by the Reuleaux triangle's construction from a regular polygon and its being, among Reuleaux polygons, the one described by the source as farthest from a ball. The source does not establish the conjecture.