The positivity conjecture for character sums f_q

Let $q$ be squarefree with $q\equiv 3 \pmod 8$, and let $f_q(x)$ denote the function defined earlier in the paper. Positivity conjecture. One has

$$f_q(x)\geqslant 0$$

for $0\leqslant x\leqslant 1/2$. This is presented as a surprising extension of the verified case $q=163$ and related computations; no proof or resolution is supplied.