Dimension of Specht Modules $S^\lambda$

The opposite is true. It is a result of D. Craven, settling a conjecture of A. Moreto, that given any $k$, for all large enough $n$, there are at least $k$ distinct irreducible representations of $S_n$ all of the same dimension.

http://arxiv.org/pdf/0709.0897.pdf

An even stronger counter-example is presented in Example 8 in my paper. It shows that $\lambda = (8,5,4)$ and $\mu=(7,7,2,1)$ both have the same multi-set of hook values. This of course implies $f^\lambda=f^\mu$.

What is interesting though is that the corresponding Ehrhart functions for the posets defined by $\lambda$ and $\mu$ are different.

Perhaps you can salvage your conjecture, and ask if for $|\lambda|>8$, the Ehrhart polynomial is unique (up to transposition). The value of $f^\lambda$ can be recovered from the leading coefficient of the Ehrhart polynomial.