The logarithmic Burgess-like character-sum conjecture

Let $q$ be a positive integer, let $\chi$ be a non-principal character modulo $q$, and let $x$ satisfy

$$\frac{\log x}{\log \log q}\longrightarrow\infty.$$

Burgess-like character-sum conjecture. For a fixed $\epsilon>0$, one has

$$\sum_{n\leqslant x}\chi(n)\ll_{\epsilon} \frac{x}{(\log x)^{3+\epsilon}}.$$

This is the Burgess-like estimate used to control the relevant exponential character sums; the paper notes that this bound is proven assuming the generalized Riemann hypothesis, while the conjectural formulation is used to derive the subsequent estimate.