High rank torus actions on contact manifolds

We prove LeBrun–Salamon conjecture in the following situation: if X is a contact Fano manifold of dimension 2n+1 whose group of automorphisms is reductive of rank ≥max(2,(n−3)/2) then X is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.