On the existence of balanced and SKT metrics on nilmanifolds

On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily K"ahler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with J, then (M, J) is necessarily K"ahler. We show that the conjecture is true for nilmanifolds.