The fraction of an $S_n$-orbit on a hyperplane

Huang, McKinnon, and Satriano conjectured that if $v \in \mathbb{R}^n$ has distinct coordinates and $n \geq 3$, then a hyperplane through the origin other than $\sum_i x_i = 0$ contains at most $2\lfloor n/2 \rfloor (n-2)!$ of the vectors obtained by permuting the coordinates of $v$. We prove this conjecture.