The finite-or-continuum conjecture for minimal coloring counts

A finite system $\mathcal{L}$ of linear homogeneous equations over $\mathbb{Q}$ is called nonregular if it is not regular over $\mathbb{Q}$. Let $\Delta(\mathcal{L};\mathbb{Q})$ denote the number of minimal colorings, up to isomorphism, of the nonzero rational numbers.

The finite-or-continuum conjecture. For every nonregular finite system $\mathcal{L}$ of linear homogeneous equations, either $\Delta(\mathcal{L};\mathbb{Q})$ is finite or $2^{\aleph_0}$. In particular, there is no $\mathcal{L}$ satisfying

$$\Delta(\mathcal{L};\mathbb{Q})=\aleph_0.$$

The paper establishes examples with finitely many and with $2^{\aleph_0}$ minimal colorings, including the equation $x_1+x_2+x_3=4x_4$. Whether countably infinite values can occur remains open.