Laugesen–Polterovich conjecture for the planar convex Neumann eigenvalue

Let $\Omega\subset\mathbb{R}^2$ be a planar convex domain, let $P(\Omega)$ denote its perimeter, and let $\mu_1(\Omega)$ denote the second Neumann eigenvalue of the Laplacian. Laugesen–Polterovich conjecture. For every planar convex domain,

$$P^2(\Omega)\mu_1(\Omega)\leq 16\pi^2.$$

Equality is achieved by squares and equilateral triangles. The problem asks for the maximizer of the second Neumann eigenvalue under a perimeter constraint; existence is known among convex sets, but the convex maximizer was not known in general when this conjecture was stated.