The square-root bound conjecture for Dedekind sums at Eisenstein norms

Let $\delta$ be a given square-free integer. Let $f>3$ run over odd integers of the form

$$f=a^2+ab+b^2,$$

with $\gcd(a,b)=1$ and $\gcd(\delta,f)=1$. For $h\in({\mathbb Z}/\delta f{\mathbb Z})^*$ satisfying $h\equiv a/b\pmod f$, consider the Dedekind sum $s(h,\delta f)$.

Square-root bound conjecture. One has

$$s(h,\delta f)=O(\sqrt f).$$

Consequently, for a given square-free integer $d_0$, the quantity in the cited proposition should satisfy $N_{d_0}(f,H)=O(\sqrt f)$ whenever $\gcd(d_0,f)=1$.

The statement is motivated by the preceding estimates for the relevant Dedekind-Rademacher sums; its general validity is not established in the supplied text.