Arithmetic pattern conjecture

Let $k$ and $N$ be positive integers. A $k$-term arithmetic progression has a basepoint $x$ and a common difference. Let $G_k(N)$ be the size of the smallest set of integers containing such a progression with basepoint $x$ for every $x\in[N]$, and let $G'_k(N)$ be the analogous minimum when the basepoints are $N$ distinct integer values. Arithmetic pattern conjecture.

$$\lim_{k\to\infty}\lim_{N\to\infty}\frac{\log G_k(N)}{\log N}=1$$

and

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

This is presented as the type-(a) dual of the arithmetic Kakeya problem and is stated to be equivalent to the arithmetic Kakeya conjecture. The source gives no resolution of either formulation.