The restricted $(U_k)$-pattern conjecture

Let $(U_k)_{k\in\mathbb{N}}$ be an increasing chain of subsets of $\mathbb{N}^+$ with $|U_k|=k$. Let $G_{U_k}(N)$ be the size of the smallest set of integers containing a $U_k$-pattern with basepoint $x$ for every $x\in[N]$. The restricted $(U_k)$-pattern conjecture. For every such chain,

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

The source states that, for chains contained in $\mathbb{N}^+$, this restricted conjecture is equivalent to the preceding pattern conjecture and hence to the relevant arithmetic Kakeya formulation. No resolution is given.