The uniqueness conjecture for the coloring c_{2,n} of E(2,n)

For a prime $p$ and positive integer $n$, define the $n$-coloring $c_{p,n}$ of $\mathbb{Q}\setminus\{0\}$ by

$$c_{p,n}(q)\equiv v_p(q)\pmod n,$$

where $v_p(q)$ is the $p$-adic valuation. For a rational number $q$ and positive integer $n$, let $E(q,n)$ be

$$x_0+qx_1+\dotsb+q^{n-2}x_{n-2}=q^{n-1}x_{n-1}.$$

Two colorings of the same set are isomorphic if one is obtained from the other by a bijection of their color sets.

The uniqueness conjecture. For $n>2$, $c_{2,n}$ is the only $n$-coloring, up to isomorphism, of the nonzero rational numbers without a monochromatic solution to $E(2,n)$.

The conjecture was verified in the paper for $n=3$ and $n=4$, and by a computer-generated proof for $n=5$ and $n=6$. It remains open in general.