Egan conjecture holds

Given a Euclidean simplex of dimension $n⩾2$ let its radii of inscribed and circumscribed spheres be $r$ and $R$, and the distance between the centers of the inscribed and circumscribed spheres be $d\mathrm{.}$ Then, $(R-nr)(R+(n-2)r)⩾d^2\mathrm{.}$