Kindler–Safra conjecture for balanced Boolean functions on the symmetric group

Let $S_n$ be the symmetric group, and let $f\colon S_n\to\{0,1\}$ be a Boolean function. Say that $f$ is $\epsilon$-close to degree $d$ if its distance from the space of degree-$d$ functions is at most $\epsilon$, where degree is defined using polynomial expressions in the variables $x_{i,j}$. Let $\mathbb{E}[f]$ denote the uniform expectation of $f$ on $S_n$. Kindler–Safra conjecture. If $f$ is $\epsilon$-close to degree $d$ and

$$\frac{1}{3}\leq \mathbb{E}[f]\leq \frac{2}{3},$$

then $f$ is $O(\epsilon)$-close to a Boolean degree-$d$ function. This is the conjectured extension of the Kindler–Safra theorem to the symmetric group in the balanced case; the corresponding result is known on several other domains, but the symmetric-group version is presented here as a conjecture.