Polygonal Ashbaugh–Benguria–Payne–Pólya–Weinberger conjecture for regular polygons

Let $P_k\subset\mathbb{R}^2$ be an open $k$-gon, and let $P_k^*\subset\mathbb{R}^d$ be an open regular $k$-gon. For $k\geq 3$, define the fundamental ratio by $\xi(D)=\lambda_2(D)/\lambda_1(D)$, where $\lambda_1(D)\leq\lambda_2(D)\leq\cdots$ are the Dirichlet–Laplacian eigenvalues of $D$. Polygonal Ashbaugh–Benguria–Payne–Pólya–Weinberger conjecture.

$$\xi(P_k)\leq\xi(P_k^*).$$

This conjecture asserts that the regular $k$-gon optimizes the fundamental ratio among $k$-gons; the general case remains open.