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.

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**EDIT** (added 12/10/2015): Incidentally, the closely related [***Soules's conjecture***][3] whose proof would yield the above permanental dominance (and which states that the largest eigenvalues of the Schur-product matrix of a given Hermitian semidefinite matrix $A$ equals the permanent of $A$), has been very recently shown to be false: [check out this explicit counterexample!][4] (please note that the author there incorrectly calls the *Soules's conjecture* the *POT conjecture*)


  [1]: http://math.ecnu.edu.cn/~zhan/papers/ZhanICCM.pdf
  [2]: http://users.monash.edu.au/~iwanless/papers/permsurveyLAA.pdf
  [3]: http://www.sciencedirect.com/science/article/pii/0024379594901171
  [4]: http://www.researchgate.net/profile/Valery_Shchesnovich/publication/280309676_The_permanent_on_top_conjecture_is_false/links/55b2c53108aed621ddfe142a.pdf