The bounded-residue-set conjecture

Let $d\geq1$ be an integer. For every prime power $q$, let $\Omega(q)\subset\mathbb Z/q\mathbb Z$ contain at most $d$ residues, and define $\Omega(b)$ for general $b$ by the Chinese remainder theorem. The bounded-residue-set conjecture. For every $\epsilon>0$, there is a constant $N(d,\epsilon)$ such that, for every integer $b$, at most $N(d,\epsilon)$ integers $n$ satisfy

$$0\le n<b^{1-\epsilon},\qquad n\in\Omega(b).$$

The source gives a weaker exponent $1/d-\epsilon$ and explicitly states the conjecture as open; it also discusses variants for polynomial congruences.