Granville–Soundararajan conjecture on character sums and smooth numbers

Let $\rho(u)$ denote the Dickman function, let $\boldsymbol{1}_{y\text{-smooth}}(n)$ indicate that every prime factor of $n$ is at most $y$, and define

$$otag \textstyle \Psi(x,y;\chi)=\sum_{\substack{n\leq x\\ n\text{ is }y\text{-smooth}}}\chi(n),\qquad \Psi(x,y;\chi_0)=\sum_{\substack{n\leq x\\ n\text{ is }y\text{-smooth}}}\chi_0(n).$$

Here $\chi$ is a non-principal character modulo $q$, and $\chi_0$ is the principal character modulo $q$.

Granville–Soundararajan conjecture. There exists a constant $A>0$ such that, for every non-principal character $\chi$ modulo $q$ and every $1\leq x\leq q$, uniformly,

$$\sum_{n\leq x}\chi(n)=\Psi(x,y;\chi)+o\bigl(\Psi(x,y;\chi_0)\bigr),$$

where

$$y=(\log q+\log^2 x)(\log_2 q)^A.$$

This conjecture asserts that character sums are asymptotically governed by their contributions from smooth numbers, strengthening the conditional estimates discussed immediately beforehand; its resolution status is not specified in the supplied text.