Conjecture on multiplicative-order scarcity and exponential-sum cancellation

Let $b\geq 2$ be an integer. For every $q$ coprime to $b$, let $O_q(b)$ be the multiplicative order of $b$ modulo $q$, and let $\mathbb{Z}_q^*$ denote the group of invertible residue classes modulo $q$. For a function $f:\mathbb{N}\to\mathbb{R}_+$, write $o_{q\to+\infty}(1)$ for a quantity tending to zero as $q\to+\infty$.

Exponential-sum and small-order conjecture. There exists a non-increasing mapping $f:\mathbb{N}\to\mathbb{R}_+$ such that:

1. For every $q\in\mathbb{N}$ with $q\wedge b=1$, if $O_q(b)\geq q^{f(q)}$, then

$$\left|\max_{p\in\mathbb{Z}_q^*}\sum_{0\leq k\leq O_q(b)-1}\exp\left(2i\pi\frac{pb^k}{q}\right)\right|\leq O_q(b)o_{q\to +\infty}(1).$$

2. One has

$$\lim_{x\to +\infty}\frac{\log\#\{1\leq q\leq x,q\wedge b=1:O_q(b)\leq q^{f(q)}\}}{\log x}=0.$$

This conjecture is intended to provide the equidistribution input needed for the paper's discussion of Mahler's problem: sufficiently large multiplicative order should yield cancellation in the associated exponential sums, while moduli with smaller order should be sparse. The source provides no resolution status.