The order-sensitive bound conjecture for Dedekind sums modulo primes

Let $p$ be prime. Let $d>1$ be odd with $d\mid p-1$, and let $h$ be an element of order $d$ in $({\mathbb Z}/p{\mathbb Z})^*$.

Order-sensitive Dedekind-sum conjecture. There exists an absolute constant $C>0$ such that

$$\left|s(h,p)\right|\leq C p^{1-\frac{1}{\phi(d)}}.$$

This is proposed as a stronger version of the paper's individual Dedekind-sum bound, based on numerical computations. The preceding explicit calculation supports the exponent in special families, but the general assertion remains open in the supplied text.