The commonness criterion for linear equations over intervals

Let $a_1,\dots,a_k\in\mathbb{Z}$ and consider the equation

$$a_1x_1+\cdots+a_kx_k=0.$$

A canceling partition is a partition of the coefficients into pairs $\{a_i,a_j\}$ such that $a_i+a_j=0$. Call the equation common over $[n]$ if every red-blue coloring of $[n]$ has asymptotically at least as many monochromatic solutions as a uniformly random coloring. Commonness criterion conjecture. An equation is common over $[n]$ if and only if $k$ is even and has a canceling partition. The source contrasts this expected interval classification with a known classification over abelian groups whose order is relatively prime to every coefficient; the interval statement remains presented as an expectation.