The character sum cancellation conjecture for rough integers

Let $q$ be large and let $\chi$ be a non-principal character modulo $q$. For real numbers $x$ and $y$ satisfying

$$\log q\leq y\leq x,$$

and writing $P(n)$ for the largest prime factor of $n$ and $e(\alpha)=e^{2\pi i\alpha}$, the rough character sum conjecture.

$$\max_{\alpha\in[0,1]}\left|\sum_{\substack{n\leq x\P(n)>y}}\frac{\chi(n)}{n}e(\alpha n)\right|\ll 1.$$

If true, this would make the inequalities in the paper's main lower bounds equalities, supporting the claim that those results are best possible; the source gives no resolution of the conjecture.