The positive-density balanceable-set conjecture

A set $B\subseteq\mathbb N$ is balanceable if there are a constant $C$ and a $2$-coloring of $\mathbb N$ such that every cascade whose length belongs to $B$ is $C$-balanced. The positive-density balanceable-set conjecture. There exists a balanceable set of positive density. Infinite examples of density zero are known, while the positive-density case remains open.