Short character-sum conjecture

Let $n\geq 3$, let $\epsilon>0$, let $M$ be an integer, let $Q\geq 3$ be an integer, let $N\leq Q^{1/n}$ be an integer, and let $\chi$ be a real non-principal character of modulus $q\leq Q$. Short character-sum conjecture. There exists $\delta(n)>0$ such that there is a constant $C(n,\epsilon)>0$ with

$$\left|\sum_{M<m\leq M+N}\chi(m)\right|\leq C(n,\epsilon)Q^{(1-\delta(n))/n+\epsilon}.$$

This conjecture, appearing in earlier work cited by the paper, is used for estimates of sums of type I and is assumed in the paper's main results; no resolution is supplied in the source.