Conjecture C_n on short cubic character sums

Let $n\geq 3$, $Q\geq 3$, and let $N\leq Q^{1/n}$. For a non-principal cubic Dirichlet character $\chi$ of modulus $q\leq Q$, define the incomplete character sum

$$S_{\chi}(M,N):=\sum_{M<a\leq M+N}\chi(a).$$

Conjecture $C_n$. For every $\varepsilon>0$, there exists $\delta=\delta(n)>0$ such that, for all $M$,

$$S_{\chi}(M,N)\ll_{\varepsilon,n}Q^{\frac{1-\delta}{n}+\varepsilon}.$$

The implied constant depends only on $\varepsilon$ and $n$. This is a standard cancellation conjecture for short character sums, stated as similar to an earlier bound; the source gives no resolution status.