Pólya–Szegő conjecture for the fundamental tone of regular polygons

Let $n\geq 3$. An $n$-gon is a polygon with $n$ sides, and let $\lambda_1$ denote its first Dirichlet eigenvalue. Among all $n$-gons with fixed area, the regular $n$-gon is the polygon under consideration. Pólya–Szegő conjecture. For each $n\geq 3$, the regular $n$-gon uniquely minimizes $\lambda_1$ among all $n$-gons with fixed area. This conjecture concerns whether the symmetry of the regular polygon is uniquely distinguished by its fundamental tone. It is known for $n=3,4$, by work of Pólya and Szegő, while the cases $n>4$ remain open.