Polygonal generalization of the equilateral-triangle eigenvalue inequalities

Let $P(n)$ denote a polygon with $n$ sides and let $R(n)$ denote a regular polygon with $n$ sides. Write $A$ for area, $R$ for inradius, and $\lambda_1<\lambda_2$ for the first two Dirichlet eigenvalues of the Laplacian. Polygonal eigenvalue conjecture. The following inequalities should hold:

$$\lambda_1 A\big|_{P(n)}\geq \lambda_1 A\big|_{R(n)},$$ $$\lambda_1R^2\big|_{P(n)}\leq\lambda_1R^2\big|_{R(n)},$$ $$(\lambda_2-\lambda_1)R^2\big|_{P(n)}\leq(\lambda_2-\lambda_1)R^2\big|_{R(n)},$$ $$\left.\frac{\lambda_2}{\lambda_1}\right|_{P(n)}\leq\left.\frac{\lambda_2}{\lambda_1}\right|_{R(n)}.$$

This proposes a common generalization of the corresponding inequalities for the equilateral triangle, including the first-eigenvalue, spectral-gap, and eigenvalue-ratio bounds. The source does not state a resolution of the general polygonal claim; the eigenvalue-ratio inequality had only been proved there under an acuteness assumption for triangles, and the authors believed it should hold for all triangles.