Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals

We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace--Neumann eigenvalue. This is the quadrilateral case of the Neumann analogue of the long-standing Pólya--Szegő conjecture, which asserts that the regular $n$-gon minimizes the first