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removed misclaim on my part!
Suvrit
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This question is better known as the permanental dominance conjecture 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 by Cheon and Wanless.


EDIT (added 12/10/2015): Incidentally, the closely related Soules's conjecture 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!.

Suvrit
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