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on Tableaux problem 27

The statement

27.[Stan1999a]*Ch. 7 Exer. 55(b) Let $\rho^{\lambda}:\mathfrak{S}_{n}\rightarrow\operatorname{GL}(m,\mathbb{C})$ be an irreducible representation of $\mathfrak{S}_{n}$ with character $\chi^{\lambda}$ (so $m=f^{\lambda}$ ). Is it possible to count the number of $\lambda$ ’s for which $\rho^{\lambda}(\mathfrak{S}_{n})\subset\operatorname{SL}(m,\mathbb{C})$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.