This question is better known as the [**permanental dominance conjecture**][1] and is still an open problem.

According to Zhan's survey, it has been confirmed for every irreducible character  of $S_n$ for $n \le 13$. Another reference cited for this conjecture is [this survey on open problems about permanents][2] by Cheon and Wanless.


  [1]: http://math.ecnu.edu.cn/~zhan/papers/ZhanICCM.pdf
  [2]: http://users.monash.edu.au/~iwanless/papers/permsurveyLAA.pdf