On a rigidity result for the first conformal eigenvalue of the Laplacian

Given $(M,g)$ a smooth compact Riemannian manifold without boundary of dimension $n\geq 3$, we consider the first conformal eigenvalue which is by definition the supremum of the first eigenvalue of the Laplacian among all metrics conformal to $g$ of volume 1. We prove that it is always greater than $n\omega_n^{\frac{2}{n}}$, the value it takes in the conformal class of the round sphere, except if $(M,g)$ is conformally diffeomorphic to the standard sphere.