Farmer–Gonek–Hughes conjecture on character sums

Let $\chi$ be a primitive character modulo $q$, and write $\tau=|t|+2$.

Farmer–Gonek–Hughes conjecture. As $q\to\infty$, there is a constant $B$ such that

$$|S(t,\chi)| \leq \bigl(B+o(1)\bigr)\sqrt{\log q\log\log q}+O(\log\tau).$$

The conjecture would substantially improve the currently admissible growth for $|S(t,\chi)|$ under GRH and could improve the paper's bounds for least character non-residues. The source attributes it to heuristics and predictions of Farmer, Gonek, and Hughes; it does not state that the conjecture has been proved.