nt.number theory – In the sum \$ sum _ { pi in S_n} e ^ {2 pi i sum_ {k = 1} ^ nk pi (k) / n} \$ (II)

In question 316836, I introduced the new sum
$$S (n) = sum_ { pi in S_n} e ^ {2 pi i sum_ {k = 1} ^ nk pi (k) / n} = text {by}[e^{2pi i jk/n}]_ {1 le j, k le n}$$
and proved that $$S (2n) = 0$$ for all $$n = 1,2,3, ldots$$. I also raised a conjecture about $$S (2n + 1)$$ which was confirmed by Noam D. Elkies and Gjergji Zaimi in the two responses there.

I have computed $$S (2n + 1)$$ for all $$n = 0.1, ldots, 8$$ and I found that
$$begin {gather} S (1) = 1, S (3) = – 3, S (5) = – 5, S (7) = – 105, S (9) = 81, \ S (11) = 6765, S (13) = 175747, S (15) = 30375, S (17) = 25219857. End {meet}$$
In view of these data, I pose the following new conjecture.

Guess. $$S (2n + 1)> 0$$ for all $$n = 4.5, ldots$$. Further, $$S (n) ge n ^ 2$$ for all $$n = 9,11,13, ldots$$.