  Using our earlier results on polynomiality properties of plethystic
logarithms of generating series of certain type we show that Schiffmann's
formulas for various counts of Higgs bundles over finite fields can be reduced
to much simpler formulas conjectured by Mozgovoy. In particular, our result
implies the conjecture of Hausel and Rodriguez-Villegas on the Poincar\'e
polynomials of twisted character varieties and the conjecture of Hausel and
Thaddeus on independence of $E$-polynomials on the degree.
