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.

---
**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]. (If the first link does not work, try [this link on Dropbox][5])


  [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/280495158_The_permanent_on_top_conjecture_is_false/links/55b6c96108aed621de043bdd.pdf?inViewer=true&pdfJsDownload=true&disableCoverPage=true&origin=publication_detail
  [5]: https://www.dropbox.com/s/9743cosw7ouj2ma/CounterExample2POT.pdf?dl=0