Henrot–Michetti conjecture on the Neumann–Steklov eigenvalue ratio for convex planar domains

Let $\Omega\subset\mathbb{R}^2$ be a bounded, open, convex set, and let

$$F(\Omega)=\frac{|\Omega|\mu_1(\Omega)}{P(\Omega)\sigma_1(\Omega)},$$

where $\mu_1(\Omega)$ and $\sigma_1(\Omega)$ are respectively the first nonzero Neumann and Steklov eigenvalues. Let $\mathcal{K}_c$ denote the class of such domains. Henrot–Michetti conjecture. For every $\Omega\in\mathcal{K}_c$,

$$1<F(\Omega)<2.$$

Moreover, the bounds are sharp: there exist sequences $R_n$ of thinning rectangles and $T_n$ of thinning triangles such that

$$\lim_{n\to\infty}F(R_n)=1,\qquad \lim_{n\to\infty}F(T_n)=2.$$

The conjecture seeks sharp universal comparison bounds between the first Neumann and Steklov eigenvalues for convex planar domains; the supplied source reports that the sharpness is supported by numerical simulations, while no resolution is given.