The general Burgess-like character-sum conjecture

Let $q$ be a positive integer, let $\chi$ be a non-principal character modulo $q$, and let $x$ satisfy $x\gg l(q)$, where $l:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is fixed. Let $c:\mathbb{N}\rightarrow\mathbb{N}$ be a function.

General Burgess-like character-sum conjecture. For a certain function $c:\mathbb{N}\rightarrow\mathbb{N}$, one has

$$\sum_{n\leqslant x}\chi(n)\ll \frac{x}{c(x)}.$$

This is a general formulation of the Burgess-like input used in the paper's exponential-sum argument. The statement leaves the functions $l$ and $c$ at the level specified above, so the strength and scope of the resulting character-sum estimate depend on their choice.