Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let $$S(n)=\bigg\{\sum_{k=1}^nk^2\pi(k)^2:\ \pi\in S_n\}.$$ Motivated by Question 316142 of mine, here I ask the following question.

QUESTION 1: Is it true that for each integer $n>4$ the set $S(n)$ contains a complete system of residues modulo $2n+1$?

I conjecture that this question has a positive answer, and I have verified this for all $n=5,6,\ldots,11$.

If $p=2n+1$ is an odd prime, then the list $1^2,2^2\ldots,n^2$ gives all the $n=(p-1)/2$ quadratic residues modulo $p$. In view of this, I also formulate the following conjecture on finite fields.

**Conjecture**. Let $\mathbb F_q$ be a finite field of order $q$ with $\text{ch}(\mathbb F_q)>3$. Let $a_1,\ldots,a_{(q-1)/2}$ be all the $(q-1)/2$ nonzero squares in $\mathbb F_q$. Then
$$\bigg\{\sum_{k=1}^{(q-1)/2} a_ka_{\pi(k)}:\ \pi\in S_{(q-1)/2}\bigg\}=\mathbb F_q.$$

QUESTION 2：Is my above conjecture for finite fields correct?

For the finite field $\mathbb F_9=\mathbb Z_3[x]/(x^2+1)$, the nonzero squares in $\mathbb F_9$ are $a_1=1,\ a_2=-1,\ a_3=x$ and $a_4=-x$. Note that $$\bigg\{\sum_{k=1}^4a_ka_{\pi(k)}:\ \pi\in S_4\bigg\}=\{0,\pm1,\pm x\}\not=\mathbb F_9.$$