Laugesen's perimeter-normalized Neumann eigenvalue conjecture for convex planar domains

Let $\Omega$ be a convex domain in $\mathbb{R}^2$, let $L(\Omega)$ denote its perimeter, and let $\mu_2(\Omega)$ be its first nonzero Neumann eigenvalue. Laugesen's conjecture.

$$L(\Omega)^2\,\mu_2(\Omega)\le 16\pi^2.$$

Equality is attained only for squares and equilateral triangles.

This conjecture asks for the sharp perimeter-normalized upper bound for the first nonzero Neumann eigenvalue among convex planar domains. It was verified by A. Raiko for parallelograms subject to certain geometric restrictions, but the general convex-domain statement remains unresolved.