The rainbow homogeneous-progression coloring conjecture

For $s,k\in\mathbb N$, let $A_{s,k}=\{s,2s,\ldots,ks\}$. A coloring is rainbow on a set when all elements of that set receive distinct colors. Rainbow-coloring conjecture. For every $k\in\mathbb N$ there is a $k$-coloring of $\mathbb N$ such that every set $A_{s,k}$ is rainbow. This problem was posed independently by Pach and Pálvölgyi; the source does not give a resolution.