A stability result for balanced dictatorships in $S_n$

We prove that a balanced Boolean function on $S_n$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_n$, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on $S_n$ generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from $0$ and $1$. In contrast, [Ellis, D., Filmus, Y., Friedgut, E., A quasi-stability result for dictatorships in $S_n$, Combinatorica 35 (2015), pp. 573-618] deals with Boolean functions of expectation O(1/n) whose Fourier transform is highly concentrated on the first two irreducible representations of $S_n$. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.