The $(U_k)$-pattern conjecture

Let $(U_k)_{k\in\mathbb{N}}$ be an increasing chain of subsets of $\mathbb{Q}^+$ with $|U_k|=k$. A $U_k$-pattern is a set obtained from $U_k$ by a common scaling and translation; $G'_{U_k}(N)$ denotes the size of the smallest set of integers containing a $U_k$-pattern with basepoint $x$ for $N$ distinct integer values of $x$. The $(U_k)$-pattern conjecture. For every such increasing chain,

$$\lim_{k\to\infty}\lim_{N\to\infty}\frac{\log G'_{U_k}(N)}{\log N}=1.$$

This is technically a family of conjectures, one for each chain $(U_k)$. The source notes that integer and rational basepoints give the same quantity after clearing denominators.