Conjecture $C_\eta$ on short character sums

Let $\eta$ be a real number satisfying $0<\eta\leq 1$. A real character is a Dirichlet character taking real values, and a character is non-principal when it is not the principal character. For a character $\chi$, let $q$ denote its conductor. Conjecture $C_\eta$. There exists a real number $\delta=\delta(\eta)>0$ such that for all $\epsilon>0$ there exists a real number $C=C(\eta,\epsilon)>0$ such that for all integers $Q\geq 3$, all non-principal real characters $\chi$ of conductor $q\leq Q$, all integers $N\leq Q^\eta$, and all integers $M$, one has

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

This is a short-interval character-sum estimate used to study the distribution of residue-field degrees in the extensions constructed in the paper. It is known for $\eta>1/4$ as a consequence of Burgess's inequality, while it remains open for $\eta\leq 1/4$.