The multiplicative decomposition conjecture for minimal coloring counts

For a system $\mathcal{L}$ of linear homogeneous equations over a ring $R$, let $\Delta(\mathcal{L};R)$ denote the number of minimal colorings, up to isomorphism, of the nonzero elements of $R$. For a rational number $q$ and positive integer $n$, let $E(q,n)$ be

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

The multiplicative decomposition conjecture. If $a$, $b$, and $n$ are integers, $n>2$, $a>1$, $|b|>1$, and $\gcd(a,b)=1$, then

$$\Delta(E(a,n);\mathbb{Q})=\Delta(E(-a,n);\mathbb{Q})$$

and

$$\Delta(E(ab,n);\mathbb{Q})=\Delta(E(a/b,n);\mathbb{Q})=\Delta(E(a,n);\mathbb{Q})+\Delta(E(b,n);\mathbb{Q}).$$

This conjecture would reduce the problem of determining $\Delta(E(q,n);\mathbb{Q})$ to the case where $q$ is a prime power. The source does not report a general resolution.