Proof of a conjecture on permutations

Given a positive integer $n$, define a function on the symmetric group $S_n$ by $$F(\tau )=\sum _{k=1}^n{k}^2\tau (k{)}^2\mathrm{.}$$ Motivated by a conjecture of Zhi-Wei Sun, we investigate the residue classes attained by $F(\tau )$ modulo $2n+1$. We prove that for every integer $n>4$, the set $\{F(\tau ):\tau \in S_n\}$ contains a complete residue system modulo $2n+1$. The proof is based on a family of involutions whose values are controlled by subset sums of squares.