Monotonicity of the first Dirichlet eigenvalue of regular polygons

In this paper we prove that the first Dirichlet eigenvalue ${\lambda }_1^N$ of an $N$-sided regular polygon of fixed area is a monotonically decreasing function of $N$ for all $N\ge 3$, as well as the monotonicity of the quotients ${\displaystyle \frac{{\lambda }_1^N}{{\lambda }_1^{N+1}}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006].