The bounded multiplicative coloring conjecture

A multiplicative $2$-coloring is a multiplicative function $f:\mathbb N\to\{-1,+1\}$. The bounded multiplicative coloring conjecture. There exists a constant $C$ and a multiplicative $2$-coloring $f$ of $\mathbb N$ by colors $\{-1,+1\}$ such that

$$\sum_{i=1}^{n}f(i)\leqslant C$$

for all $n\geqslant 1$. This is a relaxed form of the divine coloring problem and remains open.