The divine coloring conjecture

Let $f$ be a coloring of $\mathbb N$ by the two colors $\{-1,+1\}$. A finite set $A=\{a_1<\cdots<a_n\}$ is divinely colored if at least half of $a_2,\dots,a_n$ have a color different from that of $a_1$. The divine coloring conjecture. There exists a $2$-coloring of $\mathbb N$ in which every cascade is divinely colored. The problem is a stronger-looking variant of discrepancy coloring and remains open.